$3\left[\begin{smallmatrix} 1 & 2 \\ 1 & 2\end{smallmatrix}\right] -\left[\begin{smallmatrix} 3 & 2 \\ 3 & 2\end{smallmatrix}\right] - 4\left[\begin{smallmatrix} 0 & 1 \\ 0 & 1\end{smallmatrix}\right]$

(2)

$2\left[\begin{smallmatrix} 2 \\ 2 \\ 1\end{smallmatrix}\right] - 3\left[\begin{smallmatrix} 0 \\ 5 \\ 6\end{smallmatrix}\right]$

(3)

$2\left[\begin{smallmatrix} 1 & 8 & 7 & - 15 \\ 1 & - 5 & - 6 & 11\end{smallmatrix}\right] -\left[\begin{smallmatrix} 5 & 24 & - 7 & - 1 \\ - 1 & - 2 & 7 & 3\end{smallmatrix}\right]$

(4)

$\mathbf{A}_{18} + \mathbf{A}_{40}$ . $\mathbf{A}_{18} = \left[\begin{smallmatrix} 1 & 2 \\ 3 & 4 \end{smallmatrix}\right], \mathbf{A}_{40} = \left[\begin{smallmatrix} 4 & -2 \\ -3 & 1 \end{smallmatrix}\right]$ .

(5)

$\mathbf{A}_{28} - \mathbf{A}_{12}$ . $\mathbf{A}_{28} = \left[\begin{smallmatrix} 1 & 3 \\ 3 & 1 \end{smallmatrix}\right], \mathbf{A}_{12} = \left[\begin{smallmatrix} 1 & 1 \\ 1 & 1 \end{smallmatrix}\right]$ .

(6)

$2\mathbf{A}_{573} - \mathbf{A}_{571}$ . $\mathbf{A}_{573} = \left[\begin{smallmatrix} 0 & 0 & 1 & 1 & 1 \\ 0 & 1 & 0 & 0 & 1 \end{smallmatrix}\right], \mathbf{A}_{571} = \left[\begin{smallmatrix} 1 & 0 & 1 & 1 & 1 \\ 0 & 1 & 1 & 1 & 1 \end{smallmatrix}\right]$ .

(7)

$\frac{1}{2}\left(\mathbf{c}_{51} + \mathbf{c}_{52}\right)$ . $\mathbf{c}_{51} = \left[\begin{smallmatrix} 1 \\ 1 \\ -1 \end{smallmatrix}\right], \mathbf{c}_{52} = \left[\begin{smallmatrix} 1 \\ -1 \\ 1 \end{smallmatrix}\right]$ .

Problem 15.3

(1)(2)(3)(4)(5)

Describe the conditions under which the following identities hold, and prove these identities $(A, B, C$ are matrices, $\alpha , \beta$ are numbers $)$ :

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(1)

$A + B = B + A$

(2)

$A + (B + C) = (A + B) + C$

(3)

$\alpha (\beta A) = (\alpha \beta ) A$

(4)

$\alpha (A + B) = \alpha A + \alpha B$

(5)

$(\alpha + \beta ) A = \alpha A + \beta A$ .

Problem 15.4

(1)(2)

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(1)

Can a row of length $m$ be multiplied by a column of height $n$ ?

(2)

Can a column of height $n$ be multiplied by a row of length $m$ ?

Problem 15.5

(1)(2)(3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16)

Compute the product of the matrices:

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(1)

$\left[\begin{smallmatrix} 2 & - 3 & 0\end{smallmatrix}\right]\left[\begin{smallmatrix} 4 \\ 3 \\ 1\end{smallmatrix}\right]$

(2)

$\left[\begin{smallmatrix} 4 \\ 3 \\ 1\end{smallmatrix}\right] \left[\begin{smallmatrix} 2 & -3 & 0 \end{smallmatrix}\right]$

(3)

$\left[\begin{smallmatrix} 1 & 1 \\ 1 & 1\end{smallmatrix}\right]\left[\begin{smallmatrix} 3 & 5 \\ 5 & 9\end{smallmatrix}\right]$

(4)

$\mathbf{A}_{1} \mathbf{A}_{12}$ . $\mathbf{A}_{1} = \left[\begin{smallmatrix} 1 & 0 \end{smallmatrix}\right], \mathbf{A}_{12} = \left[\begin{smallmatrix} 1 & 1 \\ 1 & 1 \end{smallmatrix}\right]$ .

(5)

$\mathbf{A}_{4} \mathbf{A}_{395}$ . $\mathbf{A}_{4} = \left[\begin{smallmatrix} 0 & 1 & 0 & 0 \end{smallmatrix}\right], \mathbf{A}_{395} = \left[\begin{smallmatrix} 1 & 4 & 3 \\ 0 & 3 & 2 \\ 0 & 1 & 0 \\ 0 & 2 & 1 \end{smallmatrix}\right]$ .

(6)

$\mathbf{A}_{442} \mathbf{c}_{168}$ . $\mathbf{A}_{442} = \left[\begin{smallmatrix} 3 & 3 & -4 & -3 \\ 0 & 6 & 1 & 1 \\ 5 & 4 & 2 & 1 \\ 2 & 3 & 3 & 2 \end{smallmatrix}\right], \mathbf{c}_{168} = \left[\begin{smallmatrix} 0 \\ 1 \\ 0 \\ 0 \end{smallmatrix}\right]$ .

(7)

$\mathbf{A}_{110} \mathbf{A}_{12}$ . $\mathbf{A}_{110} = \left[\begin{smallmatrix} 3 & 1 \\ 2 & 1 \\ 1 & 0 \end{smallmatrix}\right], \mathbf{A}_{12} = \left[\begin{smallmatrix} 1 & 1 \\ 1 & 1 \end{smallmatrix}\right]$ .

(8)

$\mathbf{A}_{3} \mathbf{A}_{205}$ . $\mathbf{A}_{3} = \left[\begin{smallmatrix} 1 & 1 & 1 \end{smallmatrix}\right], \mathbf{A}_{205} = \left[\begin{smallmatrix} 1 & 2 & 3 \\ 2 & 3 & 4 \\ 3 & 4 & 5 \end{smallmatrix}\right]$ .

(9)

$\mathbf{A}_{436} \mathbf{c}_{166}$ . $\mathbf{A}_{436} = \left[\begin{smallmatrix} 1 & 1 & 1 & 1 \\ 1 & 0 & 1 & 1 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 1 & 1 \end{smallmatrix}\right], \mathbf{c}_{166} = \left[\begin{smallmatrix} 1 \\ 1 \\ 1 \\ 1 \end{smallmatrix}\right]$ .

(10)

$\mathbf{A}_{601} \mathbf{A}_{602}$ . $\mathbf{A}_{601} = \left[\begin{smallmatrix} \lambda_1 & 0 & 0 & \cdots & 0 & 0 \\ 0 & \lambda_2 & 0 & \cdots & 0 & 0 \\ 0 & 0 & \lambda_3 & \cdots & 0 & 0 \\ \vdots & \vdots & \vdots & \ddots & \vdots & \vdots \\ 0 & 0 & 0 & \cdots & \lambda_{n-1} & 0 \\ 0 & 0 & 0 & \cdots & 0 & \lambda_{n} \end{smallmatrix}\right], \mathbf{A}_{602} = \left[\begin{smallmatrix} 1/\lambda_1 & 0 & 0 & \cdots & 0 & 0 \\ 0 & 1/\lambda_2 & 0 & \cdots & 0 & 0 \\ 0 & 0 & 1/\lambda_3 & \cdots & 0 & 0 \\ \vdots & \vdots & \vdots & \ddots & \vdots & \vdots \\ 0 & 0 & 0 & \cdots & 1/\lambda_{n-1} & 0 \\ 0 & 0 & 0 & \cdots & 0 & 1/\lambda_{n} \end{smallmatrix}\right]$ .

(11)

$\mathbf{A}_{601} \mathbf{A}_{605}$ . $\mathbf{A}_{601} = \left[\begin{smallmatrix} \lambda_1 & 0 & 0 & \cdots & 0 & 0 \\ 0 & \lambda_2 & 0 & \cdots & 0 & 0 \\ 0 & 0 & \lambda_3 & \cdots & 0 & 0 \\ \vdots & \vdots & \vdots & \ddots & \vdots & \vdots \\ 0 & 0 & 0 & \cdots & \lambda_{n-1} & 0 \\ 0 & 0 & 0 & \cdots & 0 & \lambda_{n} \end{smallmatrix}\right], \mathbf{A}_{605} = \left[\begin{smallmatrix} 0 & 0 & \cdots & 0 & 0 & \lambda_1 \\ 0 & 0 & \cdots & 0 & \lambda_2 & 0 \\ 0 & 0 & \cdots & \lambda_3 & 0 & 0 \\ \vdots & \vdots & \iddots & \vdots & \vdots & \vdots \\ 0 & \lambda_{n-1} & \cdots & 0 & 0 & 0 \\ \lambda_n & 0 & \cdots & 0 & 0 & 0 \end{smallmatrix}\right]$ .

(12)

$\mathbf{A}_{605} \mathbf{A}_{601}$ . $\mathbf{A}_{605} = \left[\begin{smallmatrix} 0 & 0 & \cdots & 0 & 0 & \lambda_1 \\ 0 & 0 & \cdots & 0 & \lambda_2 & 0 \\ 0 & 0 & \cdots & \lambda_3 & 0 & 0 \\ \vdots & \vdots & \iddots & \vdots & \vdots & \vdots \\ 0 & \lambda_{n-1} & \cdots & 0 & 0 & 0 \\ \lambda_n & 0 & \cdots & 0 & 0 & 0 \end{smallmatrix}\right], \mathbf{A}_{601} = \left[\begin{smallmatrix} \lambda_1 & 0 & 0 & \cdots & 0 & 0 \\ 0 & \lambda_2 & 0 & \cdots & 0 & 0 \\ 0 & 0 & \lambda_3 & \cdots & 0 & 0 \\ \vdots & \vdots & \vdots & \ddots & \vdots & \vdots \\ 0 & 0 & 0 & \cdots & \lambda_{n-1} & 0 \\ 0 & 0 & 0 & \cdots & 0 & \lambda_{n} \end{smallmatrix}\right]$ .

(13)

$\left(\mathbf{A}_{206}\right)^{2}$ . $\mathbf{A}_{206} = \left[\begin{smallmatrix} -1 & 1 & 1 \\ -5 & 21 & 17 \\ 6 & -26 & -21 \end{smallmatrix}\right]$ .

(14)

$\left(\mathbf{A}_{200}\right)^{2}$ . $\mathbf{A}_{200} = \left[\begin{smallmatrix} 0 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{smallmatrix}\right]$ .

(15)

$\left(\mathbf{A}_{617}\right)^{2}$ . $\mathbf{A}_{617} = \left[\begin{smallmatrix} 1 & 1 & 1 & \ldots & 1 & 1 \\ 0 & 1 & 1 & \ldots & 1 & 1 \\ \ldots & \ldots & \ldots & \ldots & \ldots & \ldots \\ 0 & 0 & 0 & \ldots & 1 & 1 \\ 0 & 0 & 0 & \ldots & 0 & 1 \end{smallmatrix}\right]$ .

(16)

$\left(\mathbf{A}_{645}\right)^{2}$ for $\varepsilon = e^{2\pi i / n}$ . $\mathbf{A}_{645} = \left[\begin{smallmatrix} 1 & 1 & 1 & \ldots & 1 \\ 1 & \varepsilon & \varepsilon^2 & \ldots & \varepsilon^{n-1} \\ 1 & \varepsilon^2 & \varepsilon^4 & \ldots & \varepsilon^{2(n-1)} \\ \ldots \ldots \ldots \ldots & \ldots \ldots & \ldots \ldots & \ldots \ldots \\ 1 & \varepsilon^{n-1} & \varepsilon^{2(n-1)} & \ldots & \varepsilon^{(n-1)^2} \end{smallmatrix}\right]$ .

Problem 15.6

(1)(2)(3)

What conditions must the matrices $A$ and $B$ satisfy for:

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(1)

the product $AB$ to exist

(2)

the product $BA$ to exist

(3)

both products $AB$ and $BA$ to exist?

Problem 15.7

Express the dimensions of the matrix $AB$ in terms of the dimensions of $A$ and $B$ .

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Problem 15.8

The matrices $A, C$ have dimensions $m \times n$ and $p \times q$ respectively, and the product $ABC$ exists. What are the dimensions of the matrices $B, ABC$ ?

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Problem 15.9

(1)(2)(3)(4)(5)

Verify the validity of the identity ( $A, B, C, D$ are matrices, $\alpha$ is a number):

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(1)

$\alpha (AB) = (\alpha A) B$

(2)

$(AB) C = A(BC)$

(3)

$A(B + C) = AB + AC$

(4)

$(A + B) C = AC + BC$

(5)

$A(B + C + D) = AB + AC + AD$ .

Problem 15.10

(1)(2)(3)(4)

Verify whether the product exists, and if so, compute it:

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(1)

$\left[\begin{smallmatrix} 1 & 2 \\ 3 & 4\end{smallmatrix}\right]\left[\begin{smallmatrix} 1 & 2\end{smallmatrix}\right]\left[\begin{smallmatrix} 2 \\ 1\end{smallmatrix}\right]$

(2)

$\left[\begin{smallmatrix} 2 \\ 4\end{smallmatrix}\right]\left[\begin{smallmatrix} 1 & 2\end{smallmatrix}\right]\left[\begin{smallmatrix} 2 \\ 1\end{smallmatrix}\right]$

(3)

$\left[\begin{smallmatrix} 1 \\ 2 \\ 2 \end{smallmatrix}\right]\left[\begin{smallmatrix} 2 & 1\end{smallmatrix}\right]\left[\begin{smallmatrix} 2 \\ 4 \end{smallmatrix}\right]$

(4)

$\mathbf{A}_{2} \mathbf{A}_{8} \mathbf{c}_{8} \mathbf{A}_{2}$ . $\mathbf{A}_{2} = \left[\begin{smallmatrix} -12 & 13 \end{smallmatrix}\right], \mathbf{A}_{8} = \left[\begin{smallmatrix} 13574 & 13647 \\ 28423 & 28523 \end{smallmatrix}\right], \mathbf{c}_{8} = \left[\begin{smallmatrix} -1 \\ 1 \end{smallmatrix}\right], \mathbf{A}_{2} = \left[\begin{smallmatrix} -12 & 13 \end{smallmatrix}\right]$ .

Problem 15.11

(1)(2)(3)(4)(5)(6)(7)(8)(9)(10)

Compute:

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(1)

$\left[\begin{smallmatrix} 1 & 1 \\ 1 & 1\end{smallmatrix}\right]^{n}$

(2)

$\left[\begin{smallmatrix} 1 & 1 & 1 \\ 0 & 0 & 0 \\ 0 & 0 & 0\end{smallmatrix}\right]^{n}$

(3)

$\left[\begin{smallmatrix} 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0\end{smallmatrix}\right]^{3}$

(4)

$\left(\mathbf{A}_{5}\right)^{n}$ . $\mathbf{A}_{5} = \left[\begin{smallmatrix} 1 & -1 \\ 1 & -1 \end{smallmatrix}\right]$ .

(5)

$\left(\mathbf{A}_{13}\right)^{n}$ . $\mathbf{A}_{13} = \left[\begin{smallmatrix} 1 & 1 \\ 0 & 1 \end{smallmatrix}\right]$ .

(6)

$\left(\mathbf{A}_{14}\right)^{n}$ . $\mathbf{A}_{14} = \left[\begin{smallmatrix} 1 & 0 \\ 1 & 0 \end{smallmatrix}\right]$ .

(7)

$\left(\mathbf{A}_{77}\right)^{n}$ . $\mathbf{A}_{77} = \left[\begin{smallmatrix} \cos \alpha & -\sin \alpha \\ \sin \alpha & \cos \alpha \end{smallmatrix}\right]$ .

(8)

$\left(\mathbf{A}_{601}\right)^{n}$ . $\mathbf{A}_{601} = \left[\begin{smallmatrix} \lambda_1 & 0 & 0 & \cdots & 0 & 0 \\ 0 & \lambda_2 & 0 & \cdots & 0 & 0 \\ 0 & 0 & \lambda_3 & \cdots & 0 & 0 \\ \vdots & \vdots & \vdots & \ddots & \vdots & \vdots \\ 0 & 0 & 0 & \cdots & \lambda_{n-1} & 0 \\ 0 & 0 & 0 & \cdots & 0 & \lambda_{n} \end{smallmatrix}\right]$ .

(9)

$\left(\mathbf{A}_{614}\right)^{n}$ . $\mathbf{A}_{614} = \left[\begin{smallmatrix} 0 & 1 & 0 & \ldots & 0 \\ 0 & 0 & 1 & \ldots & 0 \\ \ldots & \ldots & \ldots & \ldots & . \\ 0 & 0 & 0 & \ldots & 1 \\ 1 & 0 & 0 & \ldots & 0 \end{smallmatrix}\right]$ .

(10)

$\left(\mathbf{A}_{613}\right)^{n}$ . $\mathbf{A}_{613} = \left[\begin{smallmatrix} 0 & 1 & 0 & \ldots & 0 & 0 \\ 0 & 0 & 1 & \ldots & 0 & 0 \\ \ldots & \ldots & \ldots & \ldots & \ldots & \ldots \\ 0 & 0 & 0 & \ldots & 0 & 1 \\ 0 & 0 & 0 & \ldots & 0 & 0 \end{smallmatrix}\right]$ .

Problem 15.12

(1)(2)(3)(4)(5)(6)(7)(8)

Transpose the matrix:

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(1)

$\left[\begin{smallmatrix} 7 & - 4 \\ - 5 & 3\end{smallmatrix}\right]$

(2)

$\left[\begin{smallmatrix} 0 & & \lambda_{1} \\ \lambda_{n} & & 0\end{smallmatrix}\right]$

(3)

$\left[\begin{smallmatrix} 1 & 2 & 3\end{smallmatrix}\right]$

(4)

$\left[\begin{smallmatrix} 1 \\ 2 \\ 3 \\ 4\end{smallmatrix}\right]$

(5)

$\mathbf{A}_{9}$ . $\mathbf{A}_{9} = \left[\begin{smallmatrix} 9 & -5 \\ -5 & 3 \end{smallmatrix}\right]$ .

(6)

$\mathbf{A}_{390}$ . $\mathbf{A}_{390} = \left[\begin{smallmatrix} 2 & 1 & 1 \\ 3 & 0 & 1 \end{smallmatrix}\right]$ .

(7)

$\mathbf{A}_{544}$ . $\mathbf{A}_{544} = \left[\begin{smallmatrix} 1 & 2 & 3 & 0 & 0 \\ 2 & 3 & 4 & 0 & 0 \\ 1 & 1 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 2 \\ 0 & 0 & 0 & 3 & 4 \end{smallmatrix}\right]$ .

(8)

$\mathbf{A}_{632}$ . $\mathbf{A}_{632} = \left[\begin{smallmatrix} 0 & 1 & 1 & \ldots & 1 & 1 & 1 \\ -1 & 0 & 1 & \ldots & 1 & 1 & 1 \\ -1 & -1 & 0 & \ldots & 1 & 1 & 1 \\ \ldots & \ldots & \ldots & \ldots & \ldots & \ldots & \ldots \\ -1 & -1 & -1 & \ldots & -1 & -1 & 0 \end{smallmatrix}\right]$ .

Problem 15.13

(1)(2)(3)(4)

Verify the validity of the identity:

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(1)

$(\alpha A)^{T} = \alpha A^{T}$

(2)

$(AB)^{T} = B^{T} A^{T}$

(3)

$(ABC)^{T} = C^{T} B^{T} A^{T}$

(4)

$(A + B)^{T} = A^{T} + B^{T}$ .

Problem 15.14

Compute the matrix $P = E - \left(\mathbf{e}_{i} - \mathbf{e}_{k}\right)^{T}\left(\mathbf{e}_{i} - \mathbf{e}_{k}\right)$ ( $\mathbf{e}_{i}$ denotes the $i$ -th row of the identity matrix $E$ ).

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Problem 15.15

Let $\mathbf{a}, \mathbf{b}$ be columns of the same height and $H = \mathbf{ab}^{T}$ . Prove that $H^{2} = \lambda H$ for some number $\lambda$ .

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Problem 15.16

Is the matrix equality $AB = BA$ always true? Give examples of commuting and noncommuting matrices.

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Problem 15.17

What can be said about the dimensions of the matrices $A, B$ , if $AB = BA?$

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Problem 15.18

(1)(2)

Compute the matrix $\left[A, B\right] = AB - BA$ (the commutator of the matrices $A, B$ ), if:

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(1)

$A = \mathbf{A}_{12}, B = \mathbf{A}_{5}$ . $\mathbf{A}_{12} = \left[\begin{smallmatrix} 1 & 1 \\ 1 & 1 \end{smallmatrix}\right], \mathbf{A}_{5} = \left[\begin{smallmatrix} 1 & -1 \\ 1 & -1 \end{smallmatrix}\right]$ .

(2)

$A = \mathbf{A}_{20}, B = \mathbf{A}_{16}$ . $\mathbf{A}_{20} = \left[\begin{smallmatrix} 0 & -1 \\ 1 & 0 \end{smallmatrix}\right], \mathbf{A}_{16} = \left[\begin{smallmatrix} 1 & -1 \\ 1 & 1 \end{smallmatrix}\right]$ .

Problem 15.19

(1)(2)(3)(4)

Verify the validity of the identity (see problem 15. 18):

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(1)

$\left[A, B\right] = - \left[B, A\right]$

(2)

$\left[A, A\right] = O$

(3)

$\left[A, E\right] = \left[E, A\right] = O$

(4)

$\left[A, (B + C)\right] = \left[A, B\right] + \left[A, C\right]$

Problem 15.20

(1)(2)

Compute the matrix $\left\{ A, B\right\} = \frac{1}{2}(AB + BA)$ (the Jordan product of the matrices $A, B)$ , if:

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(1)

(2)

Problem 15.21

(1)(2)(3)(4)

Verify the validity of the identity (see problem 15.20):

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(1)

$\left\{ A, B\right\} = \left\{ B, A\right\}$

(2)

$\left\{ A, A\right\} = A^{2}$

(3)

$\left\{ A, E\right\} = A$

(4)

$\left\{ A, (B + C)\right\} = \left\{ A, B\right\} + \left\{ A, C\right\}$ .

Problem 15.22

(1)(2)(3)(4)(5)

Compute $f(A)$ , if:

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(1)

$f(t) = t^{2} - 2t + 1, A =\left[\begin{smallmatrix} 1 & 1 \\ 1 & 1\end{smallmatrix}\right]$

(2)

$f(t) = t^{2} - 2t + 1, A =\left[\begin{smallmatrix} 1 & 1 \\ 0 & 1\end{smallmatrix}\right]$

(3)

$f(t) = t^{2} - 3t + 2, A = \mathbf{A}_{11}$ . $\mathbf{A}_{11} = \left[\begin{smallmatrix} 0 & -2 \\ 1 & 3 \end{smallmatrix}\right]$ .

(4)

$f(t) = (t - \varepsilon )^{2}, A = \mathbf{A}_{78}$ . $\mathbf{A}_{78} = \left[\begin{smallmatrix} \varepsilon & 1 \\ -1 & \varepsilon \end{smallmatrix}\right]$ .

(5)

$f(t) = t^{2} + t + 1, A = \mathbf{A}_{206}$ . $\mathbf{A}_{206} = \left[\begin{smallmatrix} -1 & 1 & 1 \\ -5 & 21 & 17 \\ 6 & -26 & -21 \end{smallmatrix}\right]$ .

Problem 15.23

(1)(2)

Factoring the polynomial $f(t)$ , compute $f(A)$ , if:

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(1)

$f(t) = t^{2} - t, A = \mathbf{A}_{230}$ . $\mathbf{A}_{230} = \left[\begin{smallmatrix} -1 & 4 & 3 \\ -2 & 5 & 3 \\ 2 & -4 & -2 \end{smallmatrix}\right]$ .

(2)

$f(t) = t^{2} + 2t - 3, A = \mathbf{A}_{214}$ . $\mathbf{A}_{214} = \left[\begin{smallmatrix} 13 & 16 & 16 \\ -5 & -7 & -6 \\ -6 & -8 & -7 \end{smallmatrix}\right]$ .

Problem 15.24

(1)(2)(3)(4)

Verify whether the matrix identities hold:

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(1)

$(A + B)^{2} = A^{2} + 2AB + B^{2}$

(2)

$(A + B)(A - B) = (A - B)(A + B)$

(3)

$A^{2} - B^{2} = (A + B)(A - B)$

(4)

$(A + E)^{3} = A^{3} + 3A^{2} + 3A + E$ .

Subsection 15.2The relationship between matrix multiplication and elementary transformations (15.25-15.38)[14/71%]

Problem 15.25

Prove that the $k$ -th column of the matrix $AB$ equals the product of the matrix $A$ and the $k$ -th column of $B$ .

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Problem 15.26

State and prove a proposition analogous to 15. 25, for rows.

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Problem 15.27

Prove that the $k$ -th column of the matrix $AB$ equals a linear combination of the columns of the matrix $A$ with coefficients from the elements of the $k$ -th column of the matrix $B$ .

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Problem 15.28

State and prove the analogue of proposition 15. 27 for rows.

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Problem 15.29

(1)(2)(3)

Prove that:

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(1)

when two rows of the matrix $A$ are interchanged, the corresponding rows in $AB$ are also interchanged

(2)

if the $k$ -th row of the matrix $A$ is multiplied by a number $\lambda$ , then the $k$ -th row of $AB$ is also multiplied by $\lambda$

(3)

if the $j$ -th row of the matrix $A$ is added to its $k$ -th row, then the same elementary transformation happens to the matrix $AB$ .

Problem 15.30

(1)(2)(3)

State and prove the analogues of propositions 15.29 for columns. Namely, prove that

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(1)

when two columns of the matrix $A$ are interchanged, the corresponding columns in $AB$ are also interchanged

(2)

if the $k$ -th column of the matrix $A$ is multiplied by a number $\lambda$ , then the $k$ -th column of $AB$ is also multiplied by $\lambda$

(3)

if the $j$ -th column of the matrix $A$ is added to its $k$ -th column, then the same elementary transformation happens to the matrix $AB$ .

Problem 15.31

(1)(2)

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(1)

Prove that adding to a row of a matrix a linear combination of its remaining rows can be accomplished by successively applying basic elementary row transformations.

(2)

Prove the analogous statement for the transformation consisting of interchanging two rows of a matrix.

Problem 15.32

Compute the product $\mathbf{e}_{i} A\mathbf{e}_{k}^{T}$ for an arbitrary matrix $A$ ( $\mathbf{e}_{i}$ denotes the $i$ -th row of the identity matrix of a suitable size).

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Problem 15.33

(1)(2)

For an arbitrary matrix $A$ and a matrix unit $E_{ij}$ of a suitable size, compute the product:

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(1)

$E_{ij} A$

(2)

$AE_{ij}$ .

Problem 15.34

Let the matrices $A$ and $B$ be such that for arbitrary columns $\xi$ and $\eta$ of a suitable height the equality $\xi^{T} A\eta = \xi^{T} B\eta$ holds. Prove that $A = B$ .

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Problem 15.35

Let $A$ be a matrix of dimensions $m \times n, E_{m}$ and $E_{n}$ identity matrices of order $m$ and $n$ respectively. Prove that $E_{m} A = AE_{n} = A$ .

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Problem 15.36

(1)(2)

By what matrix should the matrix $A$ be multiplied in order to obtain:

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(1)

the first column of $A$ ;

(2)

the first row of $A$ ?

Problem 15.37

(1)(2)(3)

Choose an elementary matrix $K$ such that the matrix $KA$ is obtained from $A$ :

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(1)

by interchanging the first two rows of $A$

(2)

by adding the first row to the second

(3)

by multiplying the first row of $A$ by a number $\lambda \neq 0$ .

Problem 15.38

Choose an elementary matrix $K$ such that the product $AK$ is obtained from $A$ by means of a given elementary transformation of columns.

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Subsection 15.3Inverse matrix (15.39-15.65)[27/56%]

Problem 15.39

Give examples of singular and nonsingular matrices.

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Problem 15.40

Let $A$ be a singular matrix of second order, $m$ a natural number. Prove that there exists a number $\lambda$ such that $A^{m} = \lambda^{m - 1} A$ for all $m$ .

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Problem 15.41

Is a rectangular matrix invertible?

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Problem 15.42

Prove that if the matrix $B$ , inverse to $A$ , exists, then $\operatorname {det} A \neq 0, \operatorname {det} B \neq 0, \operatorname {det} B = (\operatorname {det} A)^{ - 1}$ .

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Problem 15.43

(1)(2)

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(1)

Prove that if $A, B, C$ are square matrices and $AB = E, AC = E$ , then $B = C$ .

(2)

Is the equality $AB = E$ possible for rectangular matrices? Does statement 1) hold for rectangular matrices?

Problem 15.44

Given a square matrix $A = \left\| a_{ij}\right\|$ . Write out the system of equations satisfied by the elements of the $j$ -th column of the matrix $A^{ - 1}$ .

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Problem 15.45

(1)(2)(3)(4)(5)(6)(7)(8)(9)(10)

Compute:

Origin: AI-translated from another node content of the same node

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(1)

$\left[\begin{smallmatrix} 3 & 5 \\ 5 & 9\end{smallmatrix}\right]^{ - 1}$

(2)

$\left[\begin{smallmatrix} 2 & - 1 & 0 \\ 0 & 2 & - 1 \\ - 1 & - 1 & 1\end{smallmatrix}\right]^{ - 1}$

(3)

$\left[\begin{smallmatrix} 0 & & \lambda_{1} \\ \lambda_{n} & & 0\end{smallmatrix}\right]^{ - 1}$

(4)

$\left(\mathbf{A}_{34}\right)^{ - 1}$ . $\mathbf{A}_{34} = \left[\begin{smallmatrix} 2 & 1 \\ 1 & 1 \end{smallmatrix}\right]$ .

(5)

$\left(\mathbf{A}_{77}\right)^{ - 1}$ . $\mathbf{A}_{77} = \left[\begin{smallmatrix} \cos \alpha & -\sin \alpha \\ \sin \alpha & \cos \alpha \end{smallmatrix}\right]$ .

(6)

$\left(\mathbf{A}_{6}\right)^{ - 1}$ . $\mathbf{A}_{6} = \left[\begin{smallmatrix} 3 & 4 \\ 5 & 7 \end{smallmatrix}\right]$ .

(7)

$\left(\mathbf{A}_{207}\right)^{ - 1}$ . $\mathbf{A}_{207} = \left[\begin{smallmatrix} 1 & 2 & 0 \\ 2 & 5 & -2 \\ 0 & -2 & 5 \end{smallmatrix}\right]$ .

(8)

$\left(\mathbf{A}_{203}\right)^{ - 1}$ . $\mathbf{A}_{203} = \left[\begin{smallmatrix} 2 & 2 & -1 \\ 2 & -1 & 2 \\ -1 & 2 & 2 \end{smallmatrix}\right]$ .

(9)

$\left(\mathbf{A}_{202}\right)^{ - 1}$ . $\mathbf{A}_{202} = \left[\begin{smallmatrix} 1 & 2 & 2 \\ 2 & 1 & -2 \\ 2 & -2 & 1 \end{smallmatrix}\right]$ .

(10)

$\left(\mathbf{A}_{227}\right)^{ - 1}$ . $\mathbf{A}_{227} = \left[\begin{smallmatrix} 1 & 1 & 1 \\ 1 & 2 & 2 \\ 2 & 3 & 4 \end{smallmatrix}\right]$ .

Problem 15.46

Prove that the matrix inverse to an elementary matrix is an elementary matrix.

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Problem 15.47

(1)(2)(3)(4)(5)(6)(7)(8)(9)(10)(11)

Compute the inverse of the given elementary matrix:

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(1)

$\left[\begin{smallmatrix} - 2 & 0 \\ 0 & 1\end{smallmatrix}\right]$

(2)

$\left[\begin{smallmatrix} 0 & 1 \\ 1 & 0\end{smallmatrix}\right]$

(3)

$\left[\begin{smallmatrix} 1 & 0 \\ 3 & 1\end{smallmatrix}\right]$

(4)

$\left[\begin{smallmatrix} 1 & - 2 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{smallmatrix}\right]$

(5)

$\left[\begin{smallmatrix} 3 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{smallmatrix}\right]$

(6)

$\left[\begin{smallmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 2 & 0 & 1\end{smallmatrix}\right]$

(7)

$\left[\begin{smallmatrix} 1 & 0 & 0 \\ 0 & - 1 & 0 \\ 0 & 0 & 1\end{smallmatrix}\right]$

(8)

$\left[\begin{smallmatrix} 1 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & 0\end{smallmatrix}\right]$

(9)

$\mathbf{A}_{13}$ . $\mathbf{A}_{13} = \left[\begin{smallmatrix} 1 & 1 \\ 0 & 1 \end{smallmatrix}\right]$ .

(10)

$\mathbf{A}_{43}$ . $\mathbf{A}_{43} = \left[\begin{smallmatrix} 1 & n \\ 0 & 1 \end{smallmatrix}\right]$ .

(11)

$\mathbf{A}_{200}$ . $\mathbf{A}_{200} = \left[\begin{smallmatrix} 0 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{smallmatrix}\right]$ .

Problem 15.48

(1)(2)(3)(4)(5)(6)

Verify whether the identity holds:

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(1)

$\left(A^{T}\right)^{ - 1} = \left(A^{ - 1}\right)^{T}$

(2)

$(\alpha A)^{ - 1} = \alpha^{ - 1} A^{ - 1}$

(3)

$(AB)^{ - 1} = B^{ - 1} A^{ - 1}$

(4)

$(ABC)^{ - 1} = C^{ - 1} B^{ - 1} A^{ - 1}$

(5)

$\left(A^{ - 1}\right)^{k} = \left(A^{k}\right)^{ - 1}$

(6)

$(A + B)^{ - 1} = A^{ - 1} + B^{ - 1}$ .

Problem 15.49

(1)(2)

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(1)

Prove that a square matrix can be reduced to the identity matrix by means of elementary row transformations if and only if it is nonsingular.

(2)

State and prove the analogous statement for elementary transformations of the columns of a matrix.

Problem 15.50

Prove that every nonsingular matrix is a product of elementary matrices.

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Problem 15.51

(1)(2)(3)(4)

Decompose the given matrix into a product of elementary matrices:

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(1)

$(\mathrm{p})\left[\begin{smallmatrix} 1 & 1 \\ 0 & - 2\end{smallmatrix}\right]$

(2)

$\left[\begin{smallmatrix} 1 & - 1 \\ 1 & 1\end{smallmatrix}\right]$

(3)

$\left[\begin{smallmatrix} 0 & - 2 \\ 1 & 3\end{smallmatrix}\right]$

(4)

$\left[\begin{smallmatrix} 1 & 0 & 0 \\ 1 & 1 & 2 \\ 1 & 1 & 3\end{smallmatrix}\right]$ .

Problem 15.52

(1)(2)

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(1)

Let $A, B$ be matrices of the same order, and let the matrix $A$ be reduced, by means of a chain of elementary row transformations, to the identity matrix $E$ . Into what matrix will the same chain of elementary transformations take the matrix $E$ ? The matrix $B$ ?

(2)

Answer the same questions for a chain of elementary column transformations of the matrix $A$ that reduce $A$ to $E$ .

Problem 15.53

(1)(2)

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(1)

Describe and justify a method for computing the matrix $A^{ - 1}$ using elementary row transformations of the matrix $\left\| BE\right\|^{\square }$ .

(2)

Describe and justify a method for computing the matrix $A^{ - 1}$ using elementary column transformations of the matrix $\left[\begin{smallmatrix} A \\ E\end{smallmatrix}\right]^{\square }$ .

Problem 15.54

(1)(2)(3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)

Compute:

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(1)

$\left[\begin{smallmatrix} 0 & 0 & - 1 & 0 \\ 0 & 0 & 0 & 1 \\ 1 & 0 & 0 & 0 \\ 0 & - 1 & 0 & 0\end{smallmatrix}\right]^{ - 1}$

(2)

$\left[\begin{smallmatrix} 1 & 1 & 1 & 0 \\ - 1 & 2 & 1 & 0 \\ 1 & 4 & 1 & 0 \\ 0 & 0 & 0 & 3\end{smallmatrix}\right]^{ - 1}$

(3)

$\left[\begin{smallmatrix} 1 & 1 & \ldots & 1 \\ 0 & 1 & \ldots & 1 \\ \ldots & \ldots & \ldots & . \\ 0 & 0 & \ldots & 1\end{smallmatrix}\right]^{ - 1}$

(4)

$\left(\mathbf{A}_{430}\right)^{ - 1}$ . $\mathbf{A}_{430} = \left[\begin{smallmatrix} 0 & 0 & 0 & -1 \\ 0 & 0 & -1 & 0 \\ 0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 \end{smallmatrix}\right]$ .

(5)

$\left(\mathbf{A}_{432}\right)^{ - 1}$ . $\mathbf{A}_{432} = \left[\begin{smallmatrix} 0 & -1 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & -1 \\ 0 & 0 & 1 & 0 \end{smallmatrix}\right]$ .

(6)

$\left(\mathbf{A}_{433}\right)^{ - 1}$ . $\mathbf{A}_{433} = \left[\begin{smallmatrix} 1 & -2 & 1 & 0 \\ 0 & 1 & -2 & 1 \\ 0 & 0 & 1 & -2 \\ 0 & 0 & 0 & 1 \end{smallmatrix}\right]$ .

(7)

$\left(\mathbf{A}_{434}\right)^{ - 1}$ . $\mathbf{A}_{434} = \left[\begin{smallmatrix} 1 & 2 & 2 & 2 \\ 2 & 1 & 2 & 2 \\ 2 & 2 & 1 & 2 \\ 2 & 2 & 2 & 1 \end{smallmatrix}\right]$ .

(8)

$\left(\mathbf{A}_{439}\right)^{ - 1}$ . $\mathbf{A}_{439} = \left[\begin{smallmatrix} 0 & 1 & 1 & 1 \\ -1 & 0 & 1 & 1 \\ -1 & -1 & 0 & 1 \\ -1 & -1 & -1 & 0 \end{smallmatrix}\right]$ .

(9)

$\left(\mathbf{A}_{601}\right)^{ - 1}$ . $\mathbf{A}_{601} = \left[\begin{smallmatrix} \lambda_1 & 0 & 0 & \cdots & 0 & 0 \\ 0 & \lambda_2 & 0 & \cdots & 0 & 0 \\ 0 & 0 & \lambda_3 & \cdots & 0 & 0 \\ \vdots & \vdots & \vdots & \ddots & \vdots & \vdots \\ 0 & 0 & 0 & \cdots & \lambda_{n-1} & 0 \\ 0 & 0 & 0 & \cdots & 0 & \lambda_{n} \end{smallmatrix}\right]$ .

(10)

$\left(\mathbf{A}_{614}\right)^{ - 1}$ . $\mathbf{A}_{614} = \left[\begin{smallmatrix} 0 & 1 & 0 & \ldots & 0 \\ 0 & 0 & 1 & \ldots & 0 \\ \ldots & \ldots & \ldots & \ldots & . \\ 0 & 0 & 0 & \ldots & 1 \\ 1 & 0 & 0 & \ldots & 0 \end{smallmatrix}\right]$ .

(11)

$\left(\mathbf{A}_{609}\right)^{ - 1}$ . $\mathbf{A}_{609} = \left[\begin{smallmatrix} 1 & -1 & 0 & \ldots & 0 & 0 \\ 0 & 1 & -1 & \ldots & 0 & 0 \\ \ldots & \ldots & \ldots & \ldots & \ldots & \ldots \\ 0 & 0 & 0 & \ldots & 1 & -1 \\ 0 & 0 & 0 & \ldots & 0 & 1 \end{smallmatrix}\right]$ .

(12)

$\left(\mathbf{A}_{608}\right)^{ - 1}$ . $\mathbf{A}_{608} = \left[\begin{smallmatrix} 1 & 1 & 0 & \ldots & 0 & 0 \\ 0 & 1 & 1 & \ldots & 0 & 0 \\ \cdots & \cdots & \cdots & \cdots & \cdots & \cdots \\ 0 & 0 & 0 & \ldots & 1 & 1 \\ 0 & 0 & 0 & \ldots & 0 & 1 \end{smallmatrix}\right]$ .

(13)

$\left(\mathbf{A}_{618}\right)^{ - 1}$ . $\mathbf{A}_{618} = \left[\begin{smallmatrix} 1 & 2 & 3 & \ldots & n \\ 0 & 1 & 2 & \ldots & n-1 \\ \ldots & \ldots & \ldots & \ldots & \ldots \\ 0 & 0 & 0 & \ldots & 1 \end{smallmatrix}\right]$ .

Problem 15.55

Let $A^{2} + A + E = O$ . Prove that the matrix $A$ is nonsingular, and indicate the simplest way to compute $A^{ - 1}$ .

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Problem 15.56

Let $A^{m} = O$ . Prove that $(E - A)^{ - 1} = E + A + \cdots + A^{m - 1}$ .

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Problem 15.57

The matrix $A$ commutes with $B$ . Prove that then $A^{ - 1}$ commutes with $B^{ - 1}$ (it is assumed that the matrices are invertible).

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Problem 15.58

Verify the formula $\left(S^{ - 1} AS\right)^{m} = S^{ - 1} A^{m} S$ .

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Problem 15.59

Let $S^{ - 1} AS = B$ and let $f(t)$ be a polynomial. Prove that $f(B) = S^{ - 1} f(A) S$ .

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Problem 15.60

Let $\mathbf{a}, \mathbf{b}$ be columns of the same height, $1 / \mu = 1 + \mathbf{b}^{T} \mathbf{a} \neq 0, B = E + \mathbf{ab}^{T}$ . Verify the validity of the equality $B^{ - 1} = E - \mu \mathbf{ab}^{T}$ .

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Problem 15.61

Let $\mathbf{a}, \mathbf{b}$ be columns of height $n, A$ an invertible matrix of order $n, 1 / \mu = 1 + \mathbf{b}^{T} A^{ - 1} \mathbf{a} \neq 0$ and $B = A + \mathbf{ab}^{T}$ . Verify the validity of the equality $B^{ - 1} = A^{ - 1} - \mu A^{ - 1} \mathbf{ab}^{T} A^{ - 1}$ .

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Problem 15.62

(1)(2)

Origin: AI-translated from another node content of the same node

Generated from: Stated according to the original source

(1)

Describe and justify a method for computing the product $A^{ - 1} B$ using elementary row transformations of the matrix $\left\| BE\right\|^{\square }$ .

(2)

Describe and justify a method for computing the product $AB^{ - 1}$ using elementary column transformations of the matrix $\left[\begin{smallmatrix} A \\ E\end{smallmatrix}\right]^{\square }$ .

Problem 15.63

(1)(2)(3)(4)(5)(6)

Compute the products of the matrices:

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(1)

$\left[\begin{smallmatrix} 1 & 1 \\ 0 & 1\end{smallmatrix}\right]^{ - 1} \cdot \left[\begin{smallmatrix} 2 & 5 \\ 1 & 3\end{smallmatrix}\right]$

(2)

$\mathbf{A}_{205} \mathbf{A}_{203}^{ - 1}$ . $\mathbf{A}_{205} = \left[\begin{smallmatrix} 1 & 2 & 3 \\ 2 & 3 & 4 \\ 3 & 4 & 5 \end{smallmatrix}\right], \mathbf{A}_{203} = \left[\begin{smallmatrix} 2 & 2 & -1 \\ 2 & -1 & 2 \\ -1 & 2 & 2 \end{smallmatrix}\right]$ .

(3)

$\mathbf{A}_{203}^{ - 1} \mathbf{A}_{205}$ . $\mathbf{A}_{203} = \left[\begin{smallmatrix} 2 & 2 & -1 \\ 2 & -1 & 2 \\ -1 & 2 & 2 \end{smallmatrix}\right], \mathbf{A}_{205} = \left[\begin{smallmatrix} 1 & 2 & 3 \\ 2 & 3 & 4 \\ 3 & 4 & 5 \end{smallmatrix}\right]$ .

(4)

$\mathbf{A}_{210}^{ - 1} \mathbf{A}_{205}$ . $\mathbf{A}_{210} = \left[\begin{smallmatrix} 0 & 1 & 3 \\ 2 & 3 & 5 \\ 3 & 5 & 7 \end{smallmatrix}\right], \mathbf{A}_{205} = \left[\begin{smallmatrix} 1 & 2 & 3 \\ 2 & 3 & 4 \\ 3 & 4 & 5 \end{smallmatrix}\right]$ .

(5)

$\mathbf{A}_{450}^{ - 1} \mathbf{A}_{431}$ . $\mathbf{A}_{450} = \left[\begin{smallmatrix} 1 & 2 & 3 & 4 \\ 0 & 1 & 2 & 3 \\ 0 & 0 & 1 & 2 \\ 0 & 0 & 0 & 1 \end{smallmatrix}\right], \mathbf{A}_{431} = \left[\begin{smallmatrix} 0 & 0 & -1 & 0 \\ 0 & 0 & 0 & 1 \\ 1 & 0 & 0 & 0 \\ 0 & -1 & 0 & 0 \end{smallmatrix}\right]$ .

(6)

$\mathbf{A}_{618} \mathbf{A}_{617}^{ - 1}$ . $\mathbf{A}_{618} = \left[\begin{smallmatrix} 1 & 2 & 3 & \ldots & n \\ 0 & 1 & 2 & \ldots & n-1 \\ \ldots & \ldots & \ldots & \ldots & \ldots \\ 0 & 0 & 0 & \ldots & 1 \end{smallmatrix}\right], \mathbf{A}_{617} = \left[\begin{smallmatrix} 1 & 1 & 1 & \ldots & 1 & 1 \\ 0 & 1 & 1 & \ldots & 1 & 1 \\ \ldots & \ldots & \ldots & \ldots & \ldots & \ldots \\ 0 & 0 & 0 & \ldots & 1 & 1 \\ 0 & 0 & 0 & \ldots & 0 & 1 \end{smallmatrix}\right]$ .

Problem 15.64

(1)(2)(3)(4)(5)

Let the matrices $A, C$ be nonsingular. Solve the matrix equation:

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(1)

$AX = O$

(2)

$AX = B$

(3)

$XA = B$

(4)

$AXC = B$

(5)

$A(X + C) = B$ .

Problem 15.65

(1)(2)(3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16)

Find the matrix $X$ from the equation:

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(1)

$\left[\begin{smallmatrix} 2 & 5 \\ 1 & 3\end{smallmatrix}\right] X =\left[\begin{smallmatrix} 2 & 1 \\ 1 & 1\end{smallmatrix}\right]$

(2)

$X\left[\begin{smallmatrix} 2 & 5 \\ 1 & 3\end{smallmatrix}\right] =\left[\begin{smallmatrix} 2 & 1 \\ 1 & 1\end{smallmatrix}\right]$

(3)

$\left[\begin{smallmatrix} 2 & 1 \\ 1 & 1\end{smallmatrix}\right] X =\left[\begin{smallmatrix} 10 \\ 17\end{smallmatrix}\right]$

(4)

$\left[\begin{smallmatrix} 1 & 2 & 0 \\ 2 & 5 & - 2 \\ 0 & - 2 & 5\end{smallmatrix}\right] X =\left[\begin{smallmatrix} 1 & 0 & 0 \\ 0 & 1 & 1 \\ 0 & 1 & 0\end{smallmatrix}\right]$

(5)

$X\left[\begin{smallmatrix} 2 & 2 & - 1 \\ 2 & - 1 & 2 \\ - 1 & 2 & 2\end{smallmatrix}\right] =\left[\begin{smallmatrix} 5 & 5 & 2 \\ 5 & 8 & - 1\end{smallmatrix}\right]$

(6)

$X\left[\begin{smallmatrix} 1 & 1 \\ 1 & 1\end{smallmatrix}\right] =\left[\begin{smallmatrix} 1 & - 1 \\ 1 & - 1\end{smallmatrix}\right]$

(7)

$\left[\begin{smallmatrix} 1 & 1 \\ 0 & 1\end{smallmatrix}\right] X = X\left[\begin{smallmatrix} 1 & 1 \\ 0 & 1\end{smallmatrix}\right]$

(8)

$\mathbf{A}_{12} X = \mathbf{A}_{5}$ . $\mathbf{A}_{12} = \left[\begin{smallmatrix} 1 & 1 \\ 1 & 1 \end{smallmatrix}\right], \mathbf{A}_{5} = \left[\begin{smallmatrix} 1 & -1 \\ 1 & -1 \end{smallmatrix}\right]$ .

(9)

$X\mathbf{A}_{12} = \mathbf{A}_{12}$ . $\mathbf{A}_{12} = \left[\begin{smallmatrix} 1 & 1 \\ 1 & 1 \end{smallmatrix}\right], \mathbf{A}_{12} = \left[\begin{smallmatrix} 1 & 1 \\ 1 & 1 \end{smallmatrix}\right]$ .

(10)

$X^{ - 1} \mathbf{A}_{12} X = \mathbf{A}_{21}$ . $\mathbf{A}_{12} = \left[\begin{smallmatrix} 1 & 1 \\ 1 & 1 \end{smallmatrix}\right], \mathbf{A}_{21} = \left[\begin{smallmatrix} 0 & 0 \\ 0 & 2 \end{smallmatrix}\right]$ .

(11)

$\mathbf{A}_{111} X = \mathbf{A}_{228}$ . $\mathbf{A}_{111} = \left[\begin{smallmatrix} 1 & 1 \\ 1 & 1 \\ 2 & 2 \end{smallmatrix}\right], \mathbf{A}_{228} = \left[\begin{smallmatrix} 1 & 2 & 4 \\ 1 & 2 & 4 \\ 2 & 4 & 8 \end{smallmatrix}\right]$ .

(12)

$\mathbf{A}_{229} X = \mathbf{A}_{116}$ . $\mathbf{A}_{229} = \left[\begin{smallmatrix} 2 & -1 & 0 \\ 0 & 2 & -1 \\ -1 & -1 & 1 \end{smallmatrix}\right], \mathbf{A}_{116} = \left[\begin{smallmatrix} 1 & 0 \\ 0 & 1 \\ -1 & -1 \end{smallmatrix}\right]$ .

(13)

$\mathbf{A}_{203} X = \mathbf{c}_{53}$ . $\mathbf{A}_{203} = \left[\begin{smallmatrix} 2 & 2 & -1 \\ 2 & -1 & 2 \\ -1 & 2 & 2 \end{smallmatrix}\right], \mathbf{c}_{53} = \left[\begin{smallmatrix} 7 \\ 1 \\ 1 \end{smallmatrix}\right]$ .

(14)

$\mathbf{A}_{126} X = \mathbf{A}_{230}$ . $\mathbf{A}_{126} = \left[\begin{smallmatrix} 3 & 2 \\ 3 & 1 \\ -2 & 0 \end{smallmatrix}\right], \mathbf{A}_{230} = \left[\begin{smallmatrix} -1 & 4 & 3 \\ -2 & 5 & 3 \\ 2 & -4 & -2 \end{smallmatrix}\right]$ .

(15)

$\mathbf{A}_{113} X = \mathbf{A}_{213}$ . $\mathbf{A}_{113} = \left[\begin{smallmatrix} 0 & 1 \\ 0 & 1 \\ 1 & 0 \end{smallmatrix}\right], \mathbf{A}_{213} = \left[\begin{smallmatrix} 1 & 1 & 1 \\ 2 & 2 & 2 \\ 3 & 3 & 3 \end{smallmatrix}\right]$ .

(16)

$X\mathbf{A}_{227} = \mathbf{A}_{229}$ . $\mathbf{A}_{227} = \left[\begin{smallmatrix} 1 & 1 & 1 \\ 1 & 2 & 2 \\ 2 & 3 & 4 \end{smallmatrix}\right], \mathbf{A}_{229} = \left[\begin{smallmatrix} 2 & -1 & 0 \\ 0 & 2 & -1 \\ -1 & -1 & 1 \end{smallmatrix}\right]$ .

Subsection 15.4Other operations with matrices and special types of matrices (15.66-15.130)[65/45%]

Problem 15.66

Let $A, B$ be diagonal matrices of the same order, $\alpha$ a number. Prove that the matrices $\alpha A, A + B, AB, BA$ are also diagonal and $AB = BA$ .

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Problem 15.67

(1)(2)

Let $A = \operatorname {diag}\left(\lambda_{1}, \ldots , \lambda_{n}\right)$ . Prove that:

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(1)

the columns of the matrix $BA$ are obtained by multiplying the columns of the matrix $B$ by the numbers $\lambda_{1}, \ldots , \lambda_{n}$

(2)

the rows of the matrix $AB$ are obtained by multiplying the rows of $B$ by the numbers $\lambda_{1}, \ldots , \lambda_{n}$ .

Problem 15.68

Let $A$ be a diagonal matrix, $f(t)$ a polynomial. Prove that then the matrix $f(A)$ is also diagonal.

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Problem 15.69

Let the matrix $A$ be diagonal, all its diagonal elements distinct, and $AB = BA$ . Prove that then the matrix $B$ is also diagonal.

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Problem 15.70

The matrix $A$ commutes with every diagonal matrix of order $n$ . Prove that $A$ is a diagonal matrix of order $n$ .

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Problem 15.71

The matrix $A$ commutes with all matrix units of order $n$ . Prove that $A$ is a scalar matrix.

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Problem 15.72

The matrix $A$ commutes with every matrix of order $n$ . Prove that $A$ is a scalar matrix.

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Problem 15.73

Find all matrices that commute with every nonsingular matrix of order $n$ .

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Problem 15.74

(1)(2)(3)(4)

Find the matrix Hermitian conjugate to the given matrix:

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(1)

$\mathbf{A}_{82}$ . $\mathbf{A}_{82} = \left[\begin{smallmatrix} i & 1 \\ -1 & i \end{smallmatrix}\right]$ .

(2)

$\mathbf{A}_{86}$ . $\mathbf{A}_{86} = \left[\begin{smallmatrix} 5 & i \\ -i & 1 \end{smallmatrix}\right]$ .

(3)

$\mathbf{A}_{89}$ . $\mathbf{A}_{89} = \left[\begin{smallmatrix} 1 & -i \\ 2+i & 1-2 i \end{smallmatrix}\right]$ .

(4)

$\mathbf{A}_{81}$ . $\mathbf{A}_{81} = \left[\begin{smallmatrix} 1+i \sqrt{2} & 3 \\ 1 & 1-i \sqrt{2} \end{smallmatrix}\right]$ .

Problem 15.75

(1)(2)(3)(4)(5)

Verify the validity of the identity:

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(1)

$(A + B)^{H} = A^{H} + B^{H}$

(2)

$(\alpha A)^{H} = \bar{\alpha } A^{H}$

(3)

$\left(A^{H}\right)^{H} = A$

(4)

$(A \cdot B)^{H} = B^{H} A^{H}$

(5)

$\left(A^{H}\right)^{ - 1} = \left(A^{ - 1}\right)^{H}$ .

Problem 15.76

(1)(2)(3)(4)(5)(6)(7)(8)(9)(10)(11)

Determine whether the given second-order matrix is diagonal, scalar, triangular, symmetric, skew-symmetric, Hermitian, skew-Hermitian, unitary, orthogonal, or a permutation matrix:

Origin: AI-translated from another node content of the same node

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(1)

$\mathbf{A}_{82}$ . $\mathbf{A}_{82} = \left[\begin{smallmatrix} i & 1 \\ -1 & i \end{smallmatrix}\right]$ .

(2)

$\mathbf{A}_{12}$ . $\mathbf{A}_{12} = \left[\begin{smallmatrix} 1 & 1 \\ 1 & 1 \end{smallmatrix}\right]$ .

(3)

$\mathbf{A}_{87}$ . $\mathbf{A}_{87} = \left[\begin{smallmatrix} 1 & 1+i \\ 1-i & 3 \end{smallmatrix}\right]$ .

(4)

$\mathbf{A}_{86}$ . $\mathbf{A}_{86} = \left[\begin{smallmatrix} 5 & i \\ -i & 1 \end{smallmatrix}\right]$ .

(5)

$\mathbf{A}_{77}$ . $\mathbf{A}_{77} = \left[\begin{smallmatrix} \cos \alpha & -\sin \alpha \\ \sin \alpha & \cos \alpha \end{smallmatrix}\right]$ .

(6)

$\mathbf{A}_{15}$ . $\mathbf{A}_{15} = \left[\begin{smallmatrix} -1 & 0 \\ 0 & 1 \end{smallmatrix}\right]$ .

(7)

$\mathbf{A}_{46}$ . $\mathbf{A}_{46} = \left[\begin{smallmatrix} -4 & 0 \\ 1 & 4 \end{smallmatrix}\right]$ .

(8)

$\mathbf{A}_{23}$ . $\mathbf{A}_{23} = \left[\begin{smallmatrix} 0 & 2 \\ -2 & 0 \end{smallmatrix}\right]$ .

(9)

$\frac{1}{\sqrt{2}} \mathbf{A}_{88}$ . $\mathbf{A}_{88} = \left[\begin{smallmatrix} 1 & i \\ i & 1 \end{smallmatrix}\right]$ .

(10)

$\mathbf{A}_{22}$ . $\mathbf{A}_{22} = \left[\begin{smallmatrix} 0 & 1 \\ 1 & 0 \end{smallmatrix}\right]$ .

(11)

$\frac{1}{\sqrt{2}} \mathbf{A}_{16}$ . $\mathbf{A}_{16} = \left[\begin{smallmatrix} 1 & -1 \\ 1 & 1 \end{smallmatrix}\right]$ .

Problem 15.77

(1)(2)(3)

Prove that:

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(1)

all diagonal elements of a skew-symmetric matrix equal 0

(2)

the diagonal elements of a Hermitian matrix are real

(3)

the diagonal elements of a skew-Hermitian matrix are imaginary.

Problem 15.78

(1)(2)

Prove that:

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(1)

if the matrix $A$ is Hermitian, then the matrix $iA$ is skew-Hermitian

(2)

if the matrix $A$ is skew-Hermitian, then $iA$ is Hermitian.

Problem 15.79

(1)(2)(3)

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(1)

Find the general form of Hermitian matrices of the second order.

(2)

Find the general form of skew-Hermitian matrices of the second order.

(3)

Indicate all permutation matrices of the second order.

Problem 15.80

Prove: if the matrix $A$ is diagonal and all its diagonal elements are nonzero, then $A^{ - 1}$ exists and is diagonal.

Origin: AI-translated from another node content of the same node

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Problem 15.81

Prove: if the matrix $A$ is upper triangular and all its diagonal elements are nonzero, then $A^{ - 1}$ exists and is upper triangular.

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Problem 15.82

Prove: if $A$ is a nonsingular symmetric matrix, then $A^{ - 1}$ is also a symmetric matrix.

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Problem 15.83

Prove: if $A$ is a nonsingular skew-symmetric matrix, then $A^{ - 1}$ is also a skew-symmetric matrix.

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Problem 15.84

Prove: if $A$ is an orthogonal matrix, then $A^{ - 1}$ exists and is orthogonal.

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Problem 15.85

Prove: if $A$ is a unitary matrix, then $A^{ - 1}$ exists and is unitary.

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Problem 15.86

Prove: if $A$ is a permutation matrix, then $A^{ - 1}$ exists and is also a permutation matrix.

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Problem 15.87

(1)(2)(3)(4)(5)

Prove that the given matrix is orthogonal and find its inverse:

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(1)

$\mathbf{A}_{77}$ . $\mathbf{A}_{77} = \left[\begin{smallmatrix} \cos \alpha & -\sin \alpha \\ \sin \alpha & \cos \alpha \end{smallmatrix}\right]$ .

(2)

$\mathbf{A}_{313}$ . $\mathbf{A}_{313} = \left[\begin{smallmatrix} 2/3 & 1/\sqrt{2} & -1/(3\sqrt{2}) \\ 1/3 & 0 & 4/(3\sqrt{2}) \\ -2/3 & 1/\sqrt{2} & 1/(3\sqrt{2}) \end{smallmatrix}\right]$ .

(3)

$\mathbf{A}_{318}$ . $\mathbf{A}_{318} = \left[\begin{smallmatrix} 1/\sqrt{10} & 1/\sqrt{2} & 2/\sqrt{10} \\ 1/\sqrt{10} & -1/\sqrt{2} & 2/\sqrt{10} \\ -2/\sqrt{5} & 0 & 1/\sqrt{5} \end{smallmatrix}\right]$ .

(4)

$\mathbf{A}_{322}$ . $\mathbf{A}_{322} = \left[\begin{smallmatrix} 1/\sqrt{2} & 1/\sqrt{3} & 1/\sqrt{6} \\ 0 & 1/\sqrt{3} & -2/\sqrt{6} \\ -1/\sqrt{2} & 1/\sqrt{3} & 1/\sqrt{6} \end{smallmatrix}\right]$ .

(5)

$\mathbf{A}_{445}$ . $\mathbf{A}_{445} = \left[\begin{smallmatrix} 1/\sqrt{10} & 1/2 & 1/2 & 2/\sqrt{10} \\ 2/\sqrt{10} & 1/2 & -1/2 & -1/\sqrt{10} \\ 1/\sqrt{10} & -1/2 & -1/2 & 2/\sqrt{10} \\ 2/\sqrt{10} & -1/2 & 1/2 & -1/\sqrt{10} \end{smallmatrix}\right]$ .

Problem 15.88

(1)(2)

Prove that the given matrix is unitary and find its inverse:

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(1)

$\mathbf{A}_{103}$ . $\mathbf{A}_{103} = \left[\begin{smallmatrix} 0 & i \\ -i & 0 \end{smallmatrix}\right]$ .

(2)

$\frac{1}{\sqrt{2}} \mathbf{A}_{488}$ . $\mathbf{A}_{488} = \left[\begin{smallmatrix} i & 0 & -i & 0 \\ 0 & i & 0 & -i \\ 0 & 1 & 0 & 1 \\ 1 & 0 & 1 & 0 \end{smallmatrix}\right]$ .

Problem 15.89

Let the matrices $A$ and $B$ be upper triangular. Express the elements of the matrix $AB$ in terms of the elements of the matrices $A$ and $B$ .

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Problem 15.90

Let the matrices $A$ and $B$ be upper triangular. Prove that the matrices $A + B$ and $AB$ are also upper triangular.

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Problem 15.91

(1)(2)(3)

Let the matrices $A$ and $B$ be symmetric. Prove that:

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(1)

$A + B$ is a symmetric matrix

(2)

$A^{k}$ is a symmetric matrix for any natural number $k$

(3)

the matrix $AB$ is symmetric if and only if the matrices $A$ and $B$ commute.

Problem 15.92

(1)(2)(3)(4)

Let the matrices $A$ and $B$ be skew-symmetric. Prove that:

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(1)

$A + B$ is a skew-symmetric matrix

(2)

$A^{k}$ is a skew-symmetric matrix for odd $k$ and a symmetric matrix for even $k$

(3)

the matrix $AB$ is symmetric if and only if $A$ and $B$ commute.

(4)

State and prove a necessary and sufficient condition for the skew-symmetry of the product of the matrices $A$ and $B$ .

Problem 15.93

Let $A$ be an arbitrary square matrix. Prove that the matrices $A + A^{T}$ and $AA^{T}$ are symmetric, and the matrix $A - A^{T}$ is skew-symmetric.

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Problem 15.94

Prove that any square matrix can be decomposed into the sum of a symmetric and a skew-symmetric matrix. Is this decomposition unique?

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Problem 15.95

(1)(2)(3)

Decompose the given matrix into the sum of a symmetric and a skew-symmetric matrix:

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(1)

$\mathbf{A}_{49}$ . $\mathbf{A}_{49} = \left[\begin{smallmatrix} 4 & -3 \\ 12 & -8 \end{smallmatrix}\right]$ .

(2)

$\mathbf{A}_{16}$ . $\mathbf{A}_{16} = \left[\begin{smallmatrix} 1 & -1 \\ 1 & 1 \end{smallmatrix}\right]$ .

(3)

$\mathbf{A}_{234}$ . $\mathbf{A}_{234} = \left[\begin{smallmatrix} 0 & 2 & 2 \\ 0 & -1 & -2 \\ 0 & 0 & -2 \end{smallmatrix}\right]$ .

Problem 15.96

Let $S$ be a nonsingular matrix and $S^{T} AS = B$ . Prove that each of the properties: symmetry, skew-symmetry — holds for the matrices $A$ and $B$ simultaneously (i.e., if it holds for the matrix $A$ , then it holds for $B$ as well, and conversely).

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Problem 15.97

Prove the statement: every real Hermitian matrix is symmetric.

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Problem 15.98

(1)(2)

Let the matrices $A$ and $B$ be Hermitian. Prove that:

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(1)

the matrix $A + B$ is Hermitian

(2)

the matrix $AB$ is Hermitian if and only if the matrices $A$ and $B$ commute.

Problem 15.99

Let $A$ be a Hermitian matrix and $A = B + iC$ , where $B$ and $C$ are real matrices. Prove that $B$ is a symmetric matrix, and $C$ is skew-symmetric.

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Problem 15.100

Prove that any square matrix can be decomposed into the sum of a Hermitian and a skew-Hermitian matrix. Is this decomposition unique?

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Problem 15.101

Prove: a real unitary matrix is orthogonal.

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Problem 15.102

Prove: if the matrices $A$ and $B$ are orthogonal, then $AB$ is orthogonal.

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Problem 15.103

Prove: if the matrices $A$ and $B$ are unitary, then $AB$ is unitary.

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Problem 15.104

Let $A$ be an orthogonal matrix. Prove that then the sum of the squares of the elements of any of its rows equals 1, and the sum of the pairwise products of the corresponding elements of two different rows equals 0. Are these properties defining?

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Problem 15.105

State and prove the properties of the columns of an orthogonal matrix, analogous to 15. 104.

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Problem 15.106

State and prove the properties of a unitary matrix, analogous to properties $15. 104, 15. 105$ of an orthogonal matrix.

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Problem 15.107

Prove that a permutation matrix is orthogonal.

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Problem 15.108

Prove that if $A$ and $B$ are permutation matrices, then $AB$ is also a permutation matrix.

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Problem 15.109

It is known that the matrix $A$ is diagonal and orthogonal. What can be said about its diagonal elements $\lambda_{i}$ ?

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Problem 15.110

The matrix $A$ is diagonal and unitary. What can be said about its diagonal elements $\lambda_{i}$ ?

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Problem 15.111

(1)(2)(3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)

Verify whether the given matrix is periodic, nilpotent, or stochastic; find the period, the index of nilpotency:

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(1)

$\mathbf{A}_{22}$ . $\mathbf{A}_{22} = \left[\begin{smallmatrix} 0 & 1 \\ 1 & 0 \end{smallmatrix}\right]$ .

(2)

$\mathbf{A}_{14}$ . $\mathbf{A}_{14} = \left[\begin{smallmatrix} 1 & 0 \\ 1 & 0 \end{smallmatrix}\right]$ .

(3)

$\frac{1}{2} \mathbf{A}_{12}$ . $\mathbf{A}_{12} = \left[\begin{smallmatrix} 1 & 1 \\ 1 & 1 \end{smallmatrix}\right]$ .

(4)

$\mathbf{A}_{5}$ . $\mathbf{A}_{5} = \left[\begin{smallmatrix} 1 & -1 \\ 1 & -1 \end{smallmatrix}\right]$ .

(5)

$\mathbf{A}_{77}$ . $\mathbf{A}_{77} = \left[\begin{smallmatrix} \cos \alpha & -\sin \alpha \\ \sin \alpha & \cos \alpha \end{smallmatrix}\right]$ .

(6)

$\mathbf{A}_{243}$ . $\mathbf{A}_{243} = \left[\begin{smallmatrix} 1 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & 0 \end{smallmatrix}\right]$ .

(7)

$\mathbf{A}_{235}$ . $\mathbf{A}_{235} = \left[\begin{smallmatrix} 1 & 2 & 1 \\ 1 & 2 & 4 \\ -1 & -2 & -3 \end{smallmatrix}\right]$ .

(8)

$\mathbf{A}_{237}$ . $\mathbf{A}_{237} = \left[\begin{smallmatrix} -1 & 1 & -2 \\ 3 & -3 & 6 \\ 2 & -2 & 4 \end{smallmatrix}\right]$ .

(9)

$\mathbf{A}_{236}$ . $\mathbf{A}_{236} = \left[\begin{smallmatrix} -2 & 5 & 3 \\ -2 & 5 & 3 \\ 2 & -5 & -3 \end{smallmatrix}\right]$ .

(10)

$\mathbf{A}_{430}$ . $\mathbf{A}_{430} = \left[\begin{smallmatrix} 0 & 0 & 0 & -1 \\ 0 & 0 & -1 & 0 \\ 0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 \end{smallmatrix}\right]$ .

(11)

$\mathbf{A}_{431}$ . $\mathbf{A}_{431} = \left[\begin{smallmatrix} 0 & 0 & -1 & 0 \\ 0 & 0 & 0 & 1 \\ 1 & 0 & 0 & 0 \\ 0 & -1 & 0 & 0 \end{smallmatrix}\right]$ .

(12)

$\mathbf{A}_{457}$ . $\mathbf{A}_{457} = \left[\begin{smallmatrix} 1 & 0 & 0 & -1 \\ 1 & 0 & 0 & -1 \\ 0 & 1 & -1 & 0 \\ 0 & 1 & -1 & 0 \end{smallmatrix}\right]$ .

(13)

$\frac{1}{7} \mathbf{A}_{434}$ . $\mathbf{A}_{434} = \left[\begin{smallmatrix} 1 & 2 & 2 & 2 \\ 2 & 1 & 2 & 2 \\ 2 & 2 & 1 & 2 \\ 2 & 2 & 2 & 1 \end{smallmatrix}\right]$ .

(14)

$\mathbf{A}_{613}$ . $\mathbf{A}_{613} = \left[\begin{smallmatrix} 0 & 1 & 0 & \ldots & 0 & 0 \\ 0 & 0 & 1 & \ldots & 0 & 0 \\ \ldots & \ldots & \ldots & \ldots & \ldots & \ldots \\ 0 & 0 & 0 & \ldots & 0 & 1 \\ 0 & 0 & 0 & \ldots & 0 & 0 \end{smallmatrix}\right]$ .

Problem 15.112

Verify the property: a nilpotent matrix is always singular, a periodic one is nonsingular.

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Problem 15.113

Verify the property: if $A$ is a nilpotent matrix of second order, then $A^{2} = O$ .

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Generated from: Stated according to the original source

Problem 15.114

Verify the property: a triangular matrix is nilpotent if and only if all its diagonal elements are zero.

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Generated from: Stated according to the original source

Problem 15.115

Verify the property: if the matrices $A, B$ are nilpotent and commute, then $A + B$ and $AB$ are nilpotent.

Origin: AI-translated from another node content of the same node

Generated from: Stated according to the original source

Problem 15.116

Verify the property: if the matrices $A$ and $B$ are periodic and commute, then $AB$ is a periodic matrix. Express some period of it in terms of the periods of the matrices $A, B$ .

Origin: AI-translated from another node content of the same node

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Problem 15.117

Verify the property: let $A^{m} + A^{m - 1} + \ldots + E = O$ . Prove that $A$ is a periodic matrix.

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Problem 15.118

Verify the property: every permutation matrix is periodic.

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Problem 15.119

Verify the property: let the matrix $A$ be simultaneously unitary and Hermitian. Prove that $A$ is periodic.

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Generated from: Stated according to the original source

Problem 15.120

Let $S$ be a nonsingular matrix and $S^{ - 1} AS = B$ . Verify that each of the properties: periodicity, nilpotency — holds for the matrices $A$ and $B$ simultaneously (i.e., if it holds for the matrix $A$ , then it holds for $B$ as well, and conversely).

Origin: AI-translated from another node content of the same node

Generated from: Stated according to the original source

Problem 15.121

Let the matrices $A$ and $B$ be nonnegative. Verify that then $A + B, AB$ are also nonnegative matrices.

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Problem 15.122

Let $I$ be a column of ones, and let the matrix $A$ be nonnegative. Prove that the condition $AI = I$ is a necessary and sufficient condition for $A$ to be stochastic.

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Problem 15.123

Prove that if the matrices $A$ and $B$ are stochastic, then the matrix $AB$ is also stochastic.

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Problem 15.124

Let the matrix $A$ be stochastic. Does $A^{ - 1}$ exist? Will $A^{ - 1}$ be stochastic, if it exists?

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Problem 15.125

In what case is a stochastic matrix orthogonal?

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Problem 15.126

(1)(2)

Prove the validity of the identity:

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(1)

$\operatorname {tr}(A + B) = \operatorname {tr} A + \operatorname {tr} B$

(2)

$\operatorname {tr} AB = \operatorname {tr} BA$ .

Problem 15.127

Let $A$ be a triangular matrix, $m$ a natural number. Compute the trace of the matrix $A^{m}$ .

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Problem 15.128

(1)(2)(3)

Let $A$ be an arbitrary matrix. Compute:

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(1)

$\operatorname {tr}\left(A^{T} A\right)$

(2)

$\operatorname {tr}\left(A^{H} A\right)$

(3)

Prove that if $\operatorname {tr}\left(A^{H} A\right) = 0$ , then $A = O$ .

Problem 15.129

Prove that if $A$ is a nilpotent matrix of second order, then $\operatorname {tr} A = 0$ .

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Problem 15.130

Prove that there do not exist matrices $A$ and $B$ such that $AB - BA = E$ .

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Subsection 15.5Block matrices (15.131-15.141)[11/82%]

Problem 15.131

Let $A$ and $B$ be block matrices of the second order. State the conditions under which these matrices can be multiplied. Prove that if the product $AB$ exists, then $(AB)^{\square } = A^{\square } B^{\square }$ .

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Problem 15.132

Let $A$ and $B$ be upper block-triangular matrices of the second order, and let the product $AB$ exist. Obtain a formula for computing the matrix $A^{\square } B^{\square }$ .

Origin: AI-translated from another node content of the same node

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Problem 15.133

(1)(2)(3)

Let $A$ be a block matrix of the second order, $B$ a block column matrix of two blocks.

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(1)

State the conditions under which the product $AB$ is defined.

(2)

Prove that if $AB$ exists, then $(AB)^{\square } = A^{\square } B^{\square }$ .

(3)

Obtain a formula for computing $A^{\square } B^{\square }$ .

Problem 15.134

(1)(2)(3)(4)

Let $A$ and $B$ be block-diagonal matrices. State the conditions under which:

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(1)

the product $AB$ is defined

(2)

$(AB)^{\square } = A^{\square } B^{\square }$

(3)

the products $AB$ and $BA$ are defined

(4)

$AB = BA$ .

Problem 15.135

Verify the validity of the identities $(A + B)^{\square } = A^{\square } + B^{\square }, (AB)^{\square } = A^{\square } B^{\square }$ for arbitrary block matrices.

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Problem 15.136

(1)(2)(3)(4)(5)(6)

Splitting the given matrices into blocks, compute the product:

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(1)

$\mathbf{A}_{430} \mathbf{A}_{431}$ . $\mathbf{A}_{430} = \left[\begin{smallmatrix} 0 & 0 & 0 & -1 \\ 0 & 0 & -1 & 0 \\ 0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 \end{smallmatrix}\right], \mathbf{A}_{431} = \left[\begin{smallmatrix} 0 & 0 & -1 & 0 \\ 0 & 0 & 0 & 1 \\ 1 & 0 & 0 & 0 \\ 0 & -1 & 0 & 0 \end{smallmatrix}\right]$ .

(2)

$\mathbf{A}_{432} \mathbf{A}_{450}$ . $\mathbf{A}_{432} = \left[\begin{smallmatrix} 0 & -1 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & -1 \\ 0 & 0 & 1 & 0 \end{smallmatrix}\right], \mathbf{A}_{450} = \left[\begin{smallmatrix} 1 & 2 & 3 & 4 \\ 0 & 1 & 2 & 3 \\ 0 & 0 & 1 & 2 \\ 0 & 0 & 0 & 1 \end{smallmatrix}\right]$ .

(3)

$\mathbf{A}_{450} \mathbf{A}_{433}$ . $\mathbf{A}_{450} = \left[\begin{smallmatrix} 1 & 2 & 3 & 4 \\ 0 & 1 & 2 & 3 \\ 0 & 0 & 1 & 2 \\ 0 & 0 & 0 & 1 \end{smallmatrix}\right], \mathbf{A}_{433} = \left[\begin{smallmatrix} 1 & -2 & 1 & 0 \\ 0 & 1 & -2 & 1 \\ 0 & 0 & 1 & -2 \\ 0 & 0 & 0 & 1 \end{smallmatrix}\right]$ .

(4)

$\mathbf{A}_{451} \mathbf{A}_{452}$ . $\mathbf{A}_{451} = \left[\begin{smallmatrix} 2 & -3 & 1 & 0 \\ -2 & 0 & 2 & 0 \\ 6 & 3 & -3 & 0 \\ 0 & 0 & 0 & 2 \end{smallmatrix}\right], \mathbf{A}_{452} = \left[\begin{smallmatrix} 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 2 \\ 0 & 0 & 0 & 3 \\ 1 & 2 & 3 & 4 \end{smallmatrix}\right]$ .

(5)

$\mathbf{A}_{436} \mathbf{A}_{437}$ . $\mathbf{A}_{436} = \left[\begin{smallmatrix} 1 & 1 & 1 & 1 \\ 1 & 0 & 1 & 1 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 1 & 1 \end{smallmatrix}\right], \mathbf{A}_{437} = \left[\begin{smallmatrix} 1 & 0 & -1 & -2 \\ -2 & -1 & 0 & 1 \\ 1 & 2 & 0 & 0 \\ 0 & -1 & 0 & 0 \end{smallmatrix}\right]$ .

(6)

$\mathbf{A}_{530} \mathbf{A}_{531}$ . $\mathbf{A}_{530} = \left[\begin{smallmatrix} 1 & 2 & 5 & 5 & 5 \\ 3 & 4 & 6 & 6 & 6 \\ 0 & 0 & 1 & 1 & 1 \\ 0 & 0 & 1 & 2 & 2 \\ 0 & 0 & 1 & 2 & 3 \end{smallmatrix}\right], \mathbf{A}_{531} = \left[\begin{smallmatrix} 1 & 2 & 0 & 0 & 0 \\ 3 & 4 & 0 & 0 & 0 \\ 0 & 0 & 1 & 1 & 1 \\ 0 & 0 & 1 & 2 & 2 \\ 0 & 0 & 1 & 2 & 3 \end{smallmatrix}\right]$ .

Problem 15.137

(1)(2)

Find the matrix $\left(H^{\square }\right)^{ - 1}$ , if $H$ is a block matrix:

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(1)

$H =\left[\begin{smallmatrix} E & A \\ O & E\end{smallmatrix}\right]$

(2)

$H =\left[\begin{smallmatrix} A & B \\ O & C\end{smallmatrix}\right]$ (the matrices $A$ and $C$ are invertible).

Problem 15.138

(1)(2)

Let $E$ be the identity matrix of order $r$ ; let $D$ be an arbitrary matrix of size $r \times s$ ; let $\mathbf{o}, \mathbf{b}, \mathbf{x}$ be columns. Solve the equation:

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(1)

$\left\| ED\right\|^{\square } \mathbf{x} = \mathbf{o}$

(2)

$\left\| ED\right\|^{\square } \mathbf{x} = \mathbf{b}$ .

Problem 15.139

(1)(2)(3)(4)(5)(6)(7)

Compute the Kronecker product of the matrices:

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(1)

$\mathbf{A}_{17} \otimes \mathbf{c}_{7}$ . $\mathbf{A}_{17} = \left[\begin{smallmatrix} 3 & 5 \\ 5 & 9 \end{smallmatrix}\right], \mathbf{c}_{7} = \left[\begin{smallmatrix} 4 \\ -2 \end{smallmatrix}\right]$ .

(2)

$\mathbf{c}_{7} \otimes \mathbf{A}_{17}$ . $\mathbf{c}_{7} = \left[\begin{smallmatrix} 4 \\ -2 \end{smallmatrix}\right], \mathbf{A}_{17} = \left[\begin{smallmatrix} 3 & 5 \\ 5 & 9 \end{smallmatrix}\right]$ .

(3)

$\mathbf{A}_{18} \otimes \mathbf{c}_{8}$ . $\mathbf{A}_{18} = \left[\begin{smallmatrix} 1 & 2 \\ 3 & 4 \end{smallmatrix}\right], \mathbf{c}_{8} = \left[\begin{smallmatrix} -1 \\ 1 \end{smallmatrix}\right]$ .

(4)

$\mathbf{c}_{8} \otimes \mathbf{A}_{18}$ . $\mathbf{c}_{8} = \left[\begin{smallmatrix} -1 \\ 1 \end{smallmatrix}\right], \mathbf{A}_{18} = \left[\begin{smallmatrix} 1 & 2 \\ 3 & 4 \end{smallmatrix}\right]$ .

(5)

$\mathbf{A}_{17} \otimes \mathbf{A}_{18}$ . $\mathbf{A}_{17} = \left[\begin{smallmatrix} 3 & 5 \\ 5 & 9 \end{smallmatrix}\right], \mathbf{A}_{18} = \left[\begin{smallmatrix} 1 & 2 \\ 3 & 4 \end{smallmatrix}\right]$ .

(6)

$\mathbf{A}_{18} \otimes \mathbf{A}_{17}$ . $\mathbf{A}_{18} = \left[\begin{smallmatrix} 1 & 2 \\ 3 & 4 \end{smallmatrix}\right], \mathbf{A}_{17} = \left[\begin{smallmatrix} 3 & 5 \\ 5 & 9 \end{smallmatrix}\right]$ .

(7)

$\mathbf{A}_{13} \otimes \mathbf{A}_{19}$ . $\mathbf{A}_{13} = \left[\begin{smallmatrix} 1 & 1 \\ 0 & 1 \end{smallmatrix}\right], \mathbf{A}_{19} = \left[\begin{smallmatrix} 1 & 1 \\ 1 & 2 \end{smallmatrix}\right]$ .

Problem 15.140

Let $\mathbf{a} = \left\| a_{1}, \ldots , a_{n}\right\| , \mathbf{b} = \left\| b_{1}, \ldots , b_{m}\right\|^{T}$ . Compute $\mathbf{a} \otimes \mathbf{b}, \mathbf{b} \otimes \mathbf{a}$ and compare with ba.

Origin: AI-translated from another node content of the same node

Generated from: Stated according to the original source

Problem 15.141

(1)(2)(3)(4)(5)(6)

Verify the validity of the identity:

Origin: AI-translated from another node content of the same node

Generated from: Stated according to the original source

(1)

$(\alpha A) \otimes B = \alpha (A\otimes B)$

(2)

$(A + B) \otimes C = A\otimes C + B\otimes C$

(3)

$A\otimes (B + C) = A\otimes B + A\otimes C$

(4)

$A\otimes (B\otimes C) = (A\otimes B) \otimes C$

(5)

$AB\otimes CD = (A\otimes C)(B\otimes D)$

(6)

$(A\otimes B)^{ - 1} = A^{ - 1} \otimes B^{ - 1}$ .