Math Problems Vault

Problem 17.1

Multiply the matrices:

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(a)

$\left(\begin{array}{cc}1 & n \\ 0 & 1\end{array}\right) \cdot \left(\begin{array}{cc}1 & m \\ 0 & 1\end{array}\right)$ ;

(b)

$\left(\begin{array}{rr}\cos \alpha & -\sin \alpha \\ \sin \alpha & \cos \alpha \end{array}\right) \cdot \left(\begin{array}{rr}\cos \beta & -\sin \beta \\ \sin \beta & \cos \beta \end{array}\right)$ ;

(c)

$\left(\begin{array}{rrr}3 & -4 & 5 \\ 2 & -3 & 1 \\ 3 & -5 & -1\end{array}\right) \cdot \left(\begin{array}{cc}3 & 29 \\ 2 & 18 \\ 0 & -3\end{array}\right)$ ;

(d)

$\left(\begin{array}{rrr}1 & 5 & 3 \\ 2 & -3 & 1\end{array}\right) \cdot \left(\begin{array}{rrr}2 & -3 & 5 \\ -1 & 4 & -2 \\ 3 & -1 & 1\end{array}\right)$ ;

(e)

$\left(\begin{array}{lll}1 & 2 & 1 \\ 3 & 1 & 3 \\ 1 & 2 & 1\end{array}\right) \cdot \left(\begin{array}{lll}1 & 3 & 1 \\ 2 & 1 & 2 \\ 1 & 3 & 1\end{array}\right)$ ;

(f)

$\left(\begin{array}{rrr}1 & -1 & 3 \\ -1 & 1 & -3 \\ 2 & -2 & 6\end{array}\right) \cdot \left(\begin{array}{rrr}1 & 5 & 2 \\ 0 & 3 & -1 \\ 2 & 1 & -1\end{array}\right)$ ;

(g)

$\left(\begin{array}{llll}1 & 2 & 0 & 0 \\ 2 & 1 & 0 & 0 \\ 0 & 0 & 1 & 3 \\ 0 & 0 & 3 & 1\end{array}\right) \cdot \left(\begin{array}{rrrr}1 & 1 & 0 & 0 \\ 1 & 1 & 0 & 0 \\ 0 & 0 & 1 & -1 \\ 0 & 0 & -1 & 1\end{array}\right)$ ;

(h)

$\left(\begin{array}{llll}1 & 1 & 0 & 0 \\ 1 & 2 & 0 & 0 \\ 0 & 0 & 3 & 1 \\ 0 & 0 & 1 & 1\end{array}\right) \cdot \left(\begin{array}{rrrr}1 & 1 & 0 & 0 \\ 1 & 3 & 0 & 0 \\ 0 & 0 & 1 & 2 \\ 0 & 0 & -3 & 1\end{array}\right)$ .

Problem 17.2

(a)(b)

Carry out the operations:

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(a)

$\left(\begin{array}{llll}3 & 0 & 2 & 0 \\ 0 & 1 & 2 & 1 \\ 2 & 3 & 0 & 0\end{array}\right) \cdot \left(\begin{array}{rrr}1 & -2 & 2 \\ 2 & -1 & 1 \\ -1 & 1 & -2 \\ 2 & 2 & -1\end{array}\right)+\left(\begin{array}{rrr}-2 & 0 & -3 \\ 0 & 6 & -3 \\ 5 & -2 & 8\end{array}\right)$ ;

(b)

$\left(\begin{array}{lll}3 & 0 & 2 \\ 0 & 1 & 3 \\ 2 & 2 & 0 \\ 0 & 1 & 0\end{array}\right) \cdot \left(\begin{array}{rrrr}1 & 2 & -1 & 2 \\ -2 & -1 & 1 & 2 \\ 2 & 1 & 1 & 2\end{array}\right)+\left(\begin{array}{rrrr}0 & -4 & 6 & 1 \\ 2 & 2 & -5 & -2 \\ 2 & -2 & 6 & 4 \\ 1 & 3 & 0 & 1\end{array}\right)$ .

Problem 17.3

(a)(b)(c)(d)

Compute:

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(a)

$\left(\begin{array}{rrr}1 & 2 & 2 \\ 2 & 1 & -2 \\ 2 & -2 & 1\end{array}\right)^{2}$ ;

(b)

$\left(\begin{array}{llll}0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0\end{array}\right)^{2}$ ;

(c)

$\left(\begin{array}{rrrr}1 & 1 & 1 & 1 \\ 1 & 1 & -1 & -1 \\ 1 & -1 & 1 & -1 \\ 1 & -1 & -1 & 1\end{array}\right)^{2}$ ;

(d)

$\left(\begin{array}{llll}0 & 1 & 0 & 0 \\ 0 & 0 & 2 & 0 \\ 0 & 0 & 0 & 3 \\ 0 & 0 & 0 & 0\end{array}\right)^{2}$ .

Problem 17.4

(a)(b)(c)

Compute:

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(a)

$\left(\begin{array}{rr}\cos \alpha & \sin \alpha \\ -\sin \alpha & \cos \alpha \end{array}\right)^{n}$ ;

(b)

$\left(\begin{array}{cc}\lambda & 1 \\ 0 & \lambda \end{array}\right)^{n}$ ;

(c)

$\left(\begin{array}{ll}2 & 1 \\ 5 & 3\end{array}\right) \cdot \left(\begin{array}{ll}1 & 0 \\ 1 & 1\end{array}\right) \cdot \left(\begin{array}{rr}3 & -1 \\ -5 & 2\end{array}\right)^{n}$ .

Problem 17.5

(a)(b)

Compute the value of the polynomial $f(x)$ at the matrix $A$ :

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(a)

$f(x)=x^{3}-2 x^{2}+1, \quad A=\left(\begin{array}{ccc}2 & 1 & 0 \\ 0 & 2 & 0 \\ 1 & 1 & 1\end{array}\right)$ ;

(b)

$f(x)=x^{3}-3 x+2, \quad A=\left(\begin{array}{lll}2 & 1 & 1 \\ 1 & 2 & 1 \\ 1 & 1 & 2\end{array}\right)$ .

Problem 17.6

Prove that if the matrices $A$ and $B$ commute, then

(A+B)^{n}=\sum _{i=0}^{n}\binom {n}{i} A^{i} B^{n-i}

Give an example of two matrices $A, B$ for which this formula fails.

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Problem 17.7

Compute the powers of the square matrix

H=\left[\begin{smallmatrix} 0 & 1 & 0 & \ldots & 0 \\ 0 & 0 & 1 & \ldots & 0 \\ \ldots & \ldots & \ldots & \ldots & \ldots \\ 0 & 0 & 0 & \ldots & 1 \\ 0 & 0 & 0 & \ldots & 0 \end{smallmatrix}\right]

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Problem 17.8

Let a square matrix

J=\left[\begin{smallmatrix} \lambda & 1 & 0 & \ldots & 0 & 0 \\ 0 & \lambda & 1 & \ldots & 0 & 0 \\ \ldots & \ldots & \ldots & \ldots & \ldots & \ldots \\ 0 & 0 & 0 & \ldots & \lambda & 1 \\ 0 & 0 & 0 & \ldots & 0 & \lambda \end{smallmatrix}\right]

of size $n$ be given. Prove that if $f(x)$ is a polynomial, then

f(J)=\left[\begin{smallmatrix} f(\lambda ) & \frac{f^{\prime }(\lambda )}{1!} & \frac{f^{\prime \prime }(\lambda )}{2!} & \ldots & \frac{f^{(n-2)}(\lambda )}{(n-2)!} & \frac{f^{(n-1)}(\lambda )}{(n-1)!} \\ 0 & f(\lambda ) & \frac{f^{\prime }(\lambda )}{1!} & \ldots & \frac{f^{(n-3)}(\lambda )}{(n-3)!} & \frac{f^{(n-2)}(\lambda )}{(n-2)!} \\ \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots & \ldots \ldots \ldots \ldots \ldots \ldots & \frac{f^{\prime }(\lambda )}{1!} \\ 0 & 0 & 0 & \ldots & f(\lambda ) & \frac{f^{\prime }(\lambda )}{1!} \\ 0 & 0 & 0 & \ldots & 0 & f(\lambda ) \end{smallmatrix}\right].

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Problem 17.9

Let $C, A$ be square matrices of the same size and let $f(x)$ be a polynomial. Prove that $f\left(C A C^{-1}\right)=C f(A) C^{-1}$ .

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Problem 17.10

(a)(b)

Compute $e^{A}$ , where:

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(a)

$A=\left(\begin{array}{rr}2 & 1 \\ -4 & -2\end{array}\right)$ ;

(b)

$A=\left(\begin{array}{lll}0 & 1 & 2 \\ 0 & 0 & 6 \\ 0 & 0 & 0\end{array}\right)$ .

Problem 17.11

(a)(b)

Compute $\ln A$ , where:

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(a)

$A=\left(\begin{array}{rr}3 & 1 \\ -4 & -1\end{array}\right)$ ;

(b)

$A=\left(\begin{array}{ccccc}1 & 1 & 0 & \ldots & 0 \\ 0 & 1 & 1 & \ldots & 0 \\ \ldots & \ldots & \ldots & \ldots & \ldots \\ 0 & 0 & 0 & \ldots & 1\end{array}\right)$ .

Problem 17.12

Let $A=\left(a_{i j}\right)$ be a matrix of size $m \times n$ . Prove that

A=\sum _{i, j} a_{i j} E_{i j}

where $E_{i j}$ are the matrix units.

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Problem 17.13

Prove that $E_{i j} E_{p q}=\delta_{j p} E_{i q}$ .

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Problem 17.14

Let $A$ be an arbitrary matrix. Compute $E_{i j} A$ .

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Problem 17.15

Let $A$ be an arbitrary matrix. Compute $A E_{i j}$ .

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Problem 17.16

Let $A$ be a square matrix such that $E_{i j} A=A E_{i j}$ for all matrix units $E_{i j}$ . Prove that $A=\lambda E$ for some scalar $\lambda$ .

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Problem 17.17

Let $A$ be a square matrix such that $E_{i i} A=A E_{i i}$ for all $i$ . Prove that the matrix $A$ is diagonal.

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Problem 17.18

Let a square matrix $A$ commute with all nonsingular matrices. Prove that $A=\lambda E$ .

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Problem 17.19

Find all matrices $A$ of order $n$ such that $\operatorname {tr} A X=0$ for any matrix $X$ of order $n$ .

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Problem 17.20

Prove that the trace of the product of two matrices does not depend on the order of the factors.

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Problem 17.21

Prove that if $C$ is a nonsingular matrix, then for any matrix $A$ of the same order $\operatorname {tr} C A C^{-1}=\operatorname {tr} A$ .

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Problem 17.22

For which $\lambda$ does the equation

[X, Y]=\lambda E

have a solution ( $[X, Y]$ is the commutator of the matrices $X$ and $Y$ )?

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Problem 17.23

(a)(b)

Prove that for any square matrices $A, B, C$ the following equalities hold:

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(a)

$[A, B C]=[A, B] C+B[A, C]$ ;

(b)

$[[A, B], C]+[[B, C], A]+[[C, A], B]=0$ .

Problem 17.24

Prove that for any matrices of order 2 the equality $\left[[A, B]^{2}, C\right]=0$ holds.

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Problem 17.25

Let $A, B, \ldots , D_{1}$ be square matrices of the same order. Express the product of the matrices

\left[\begin{smallmatrix} A & B \\ C & D \end{smallmatrix}\right] \cdot \left[\begin{smallmatrix} A_{1} & B_{1} \\ C_{1} & D_{1} \end{smallmatrix}\right]

in terms of the given matrices.

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Problem 17.26

Let $A$ be a real triangular matrix commuting with ${ }^{t} A$ . Prove that the matrix $A$ is diagonal.

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Problem 17.27

Let $A=\left(a_{i j}\right) \in \mathbf{M}_{n}(\mathbb {R})$ be a symmetric nonsingular matrix such that there exists $k<n$ for which $a_{i j}=0$ whenever $\left|i-j\right| \geqslant k$ . Suppose that $A={ }^{t} B \cdot B$ , where $B=\left(b_{i j}\right)$ is an upper triangular matrix. Prove that $b_{i j}=0$ if $j-i \geqslant k$ .

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Problem 17.28

Prove that any matrix with trace 0 is a sum of commutators of matrices with trace 0.

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Problem 17.29

For the matrix

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

find matrices $A$ and $B$ such that

[A, X]=X, \quad [A, B]=-B, \quad [X, B]=A

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