Math Problems Vault

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

?

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 $A^{\square } B^{\square }$ .

?

Problem 15.133

(1)(2)(3)

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

?

(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:

?

(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.

?

Problem 15.136

Splitting the given matrices into blocks, compute the product:

?

(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:

?

(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:

?

(1)

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

(2)

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

Problem 15.139

Compute the Kronecker product of the matrices:

?

(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.

?

Problem 15.141

Verify the validity of the identity:

?

(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)

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