Let A B denotes the tensor product of matrices Amxn andBpxq defined by the block matrixA B =a12B922Ba11Ba21 B::ami B am2 B:ain Ba2n B:amn Bmpxnq(a) Let A, B, C, D be defined as Amxn, Bpxq, Cnxk, and Dqxr.Prove that (AB)(C > D) = ACⒸ BD(b) Let Amxm and B₁xn be nonsingular matrices.- Prove that (AB) is nonsingular.- Find the inverse matrix of (AB)(c) Prove that the following equality applies for any Amxmand Bnxn square matricestrace(AB) = trace(A) * trace(B)

Answered: Let A B denotes the tensor product of…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

Let A B denotes the tensor product of matrices Amxn and
Bpxq defined by the block matrix
A B =
a12B
922B
a11B
a21 B
:
:
ami B am2 B
:
ain B
a2n B
:
amn B
mpxnq
(a) Let A, B, C, D be defined as Amxn, Bpxq, Cnxk, and Dqxr.
Prove that (AB)(C > D) = ACⒸ BD
(b) Let Amxm and B₁xn be nonsingular matrices.
- Prove that (AB) is nonsingular.
- Find the inverse matrix of (AB)
(c) Prove that the following equality applies for any Amxm
and Bnxn square matrices
trace(AB) = trace(A) * trace(B)

Expert Solution

Step 1

As per norms, we will be answering the first question. If you need an answer to others, then kindly re-post the question by specifying it.

We are giving the following.

$A \otimes B = {[\begin{matrix} a_{11} B & a_{12} B & \dots & a_{1 n} B \\ a_{21} B & a_{22} B & \dots & a_{2 n} B \\ ⋮ & ⋮ & ⋱ & ⋮ \\ a_{m 1} B & a_{m 2} B & \dots & a_{m n} B \end{matrix}]}_{m p \times n q}$ where $A_{m \times n}$ and $B_{p \times q}$ .

For (a),

We will be providing that $(A \otimes B) (C \otimes D) = A C \otimes B D$ where $A_{m \times n}$ , $B_{p \times q}$ , $C_{n \times k}$ , and $D_{q \times r}$ .