Chapter 1, Problem 1E

Use MATLAB to generate random $4\times 4$ matrices A and B. For each of the following, compute A1, A2, A3, and A4 as indicated and determine which of the matrices are equal (you can use MATLAB to test whether two matrices are equal by computing their difference).

(a) $A1=A*B,A2=B*A,A3=(A\text{'}*B\text{'})\text{'},A4=(B\text{'}*A\text{'})\text{'}$

(b) $A1=A\text{'}*B\text{'},A2=(A*B)\text{'},A3=B\text{'}*A\text{'},A4=(B*A)\text{'}$

(c) $A1=\text{inv}(A*B),A2=\text{inv}(A)*\text{inv}(B),A3=\text{inv}(B*A),A4=\text{inv}(B)*\text{inv}(A)$

(d) $A1=\text{inv}\left((A*B)\text{'}\right),A2=\text{inv}(A\text{'}*B\text{'}),A3=\text{inv}(A\text{'})*\text{inv}(B\text{'}),A4=\left(\text{inv}(A)*(B)\right)\text{'}$

To determine

Compute the given relationship using random matrix in MATLAB.

Answer to Problem 1E

The solution is

  $A1=A\times B=\left[\begin{array}{l}\text{ 2}\text{.5996 2}\text{.2639 2}\text{.1191 1}\text{.3488}\\ \text{ 1}\text{.7015 1}\text{.8701 1}\text{.5961 1}\text{.3071 }\\ \text{ 1}\text{.2013 0}\text{.9745 0}\text{.7282 0}\text{.2677}\\ \text{ 1}\text{.7911 1}\text{.5751 1}\text{.8113 1}\text{.3821}\end{array}\right]$

  $A2=B\times A=\left[\begin{array}{l}\text{ 1}\text{.6225 0}\text{.8782 1}\text{.7797 1}\text{.3582}\\ \text{ 1}\text{.0310 0}\text{.8872 1}\text{.1969 1}\text{.5245 }\\ \text{ 2}\text{.1538 1}\text{.1769 2}\text{.3803 1}\text{.8653}\\ \text{ 1}\text{.7066 0}\text{.8245 1}\text{.9126 1}\text{.6901}\end{array}\right]$

  $A3=(A\text{'}\times B\text{'})\text{'}=\left[\begin{array}{l}\text{ 1}\text{.6225 0}\text{.8782 1}\text{.7797 1}\text{.3582}\\ \text{ 1}\text{.0310 0}\text{.8872 1}\text{.1969 1}\text{.5245 }\\ \text{ 2}\text{.1538 1}\text{.1769 2}\text{.3803 1}\text{.8653}\\ \text{ 1}\text{.7066 0}\text{.8245 1}\text{.9126 1}\text{.6901}\end{array}\right]$

  $A4=(B\text{'}\times A\text{'})\text{'}=\left[\begin{array}{l}\text{ 2}\text{.5996 2}\text{.2639 2}\text{.1191 1}\text{.3488}\\ \text{ 1}\text{.7015 1}\text{.8701 1}\text{.5961 1}\text{.3071}\\ \text{ 1}\text{.2013 0}\text{.9745 0}\text{.7282 0}\text{.2677 }\\ \text{ 1}\text{.7911 1}\text{.5751 1}\text{.8113 1}\text{.3821}\end{array}\right]$

Explanation of Solution

Given:The matrix has been given

  $\begin{array}{l}A=rand(4)\\ B=rand(4)\end{array}$

Concept Used:

Given,

  $\begin{array}{l}A=rand(4)\\ B=rand(4)\end{array}$

Calculate the given relationship in MATLAB.

Program:

clc
clear
close all
A=rand(4);
B=rand(4);
A1=A*B
A2=B*A
A3=(A'*B')'
A4=(B'*A')'
A4-A1
A3-A2

Quarry:

• First, we will define the random matrix A and B.
• Calculate the given relationship.
• Show the difference between two matrices.

To determine

Compute the given relationship using random matrix in MATLAB.

Answer to Problem 1E

The solution is

  $A1=A\text{'}\times B\text{'}=\left[\begin{array}{l}\text{ 0}\text{.9385 0}\text{.4596 0}\text{.5554 0}\text{.8086}\\ \text{ 1}\text{.6385 0}\text{.9082 0}\text{.8031 0}\text{.8671 }\\ \text{ 1}\text{.8933 0}\text{.8566 1}\text{.0169 1}\text{.2156}\\ \text{ 1}\text{.3666 0}\text{.5372 0}\text{.6977 0}\text{.9226}\end{array}\right]$

  $A2=(A\times B)\text{'}=\left[\begin{array}{l}\text{ 0}\text{.9689 0}\text{.7103 1}\text{.3002 1}\text{.2689}\\ \text{ 0}\text{.4397 0}\text{.3713 0}\text{.5313 0}\text{.9618 }\\ \text{ 0}\text{.8009 0}\text{.4852 0}\text{.9644 1}\text{.4121}\\ \text{ 0}\text{.7379 0}\text{.6510 1}\text{.0142 1}\text{.4816}\end{array}\right]$

  $A3=B\text{'}\times A\text{'}=\left[\begin{array}{l}\text{ 0}\text{.9689 0}\text{.7103 1}\text{.3002 1}\text{.2689}\\ \text{ 0}\text{.4397 0}\text{.3713 0}\text{.5313 0}\text{.9618 }\\ \text{ 0}\text{.8009 0}\text{.4852 0}\text{.9644 1}\text{.4121}\\ \text{ 0}\text{.7379 0}\text{.6510 1}\text{.0142 1}\text{.4816}\end{array}\right]$

  $A4=(B\times A)\text{'}=\left[\begin{array}{l}\text{ 0}\text{.9385 0}\text{.4596 0}\text{.5554 0}\text{.8086}\\ \text{ 1}\text{.6385 0}\text{.9082 0}\text{.8031 0}\text{.8671}\\ \text{ 1}\text{.8933 0}\text{.8566 1}\text{.0169 1}\text{.2156 }\\ \text{ 1}\text{.3666 0}\text{.5372 0}\text{.6977 0}\text{.9226}\end{array}\right]$

Explanation of Solution

Given:The matrix has been given

  $\begin{array}{l}A=rand(4)\\ B=rand(4)\end{array}$

Concept Used:

Given,

  $\begin{array}{l}A=rand(4)\\ B=rand(4)\end{array}$

Calculate the given relationship in MATLAB.

Program:

clc
clear
close all
A=rand(4);
B=rand(4);
A1=A'*B'
A2=(A*B)'
A3=B'*A'
A4=(B*A)'
A4-A1
A3-A2

Quarry:

• First, we will define the random matrix A and B.
• Calculate the given relationship.
• Show the difference between two matrices.

To determine

Compute the given relationship using random matrix in MATLAB.

Answer to Problem 1E

The solution is

  $A1=inv(A\times B)=\left[\begin{array}{l}\text{ 10}\text{.8369 -30}\text{.8416 32}\text{.4916 22}\text{.5021}\\ \text{ -9}\text{.9010 -0}\text{.1579 3}\text{.0864 9}\text{.2304}\\ \text{ -7}\text{.7213 28}\text{.7573 -27}\text{.5727 -25}\text{.1680}\\ \text{ 5}\text{.7242 6}\text{.0655 -11}\text{.2386 -8}\text{.9117}\end{array}\right]$

  $A2=inv(A)\times inv(B)=\left[\begin{array}{l}\text{ -3}\text{.0174 3}\text{.3756 33}\text{.6054 -7}\text{.8791}\\ \text{ -18}\text{.5258 27}\text{.1564 44}\text{.2122 -35}\text{.9855}\\ \text{ 20}\text{.8870 -26}\text{.9653 -57}\text{.5365 36}\text{.2525}\\ \text{ 1}\text{.3366 -3}\text{.7610 -14}\text{.8918 7}\text{.5920}\end{array}\right]$

  $A3=inv(B\times A)=\left[\begin{array}{l}\text{ -3}\text{.0174 3}\text{.3756 33}\text{.6054 -7}\text{.8791}\\ \text{ -18}\text{.5258 27}\text{.1564 44}\text{.2122 -35}\text{.9855}\\ \text{ 20}\text{.8870 -26}\text{.9653 -57}\text{.5365 36}\text{.2525}\\ \text{ 1}\text{.3366 -3}\text{.7610 -14}\text{.8918 7}\text{.5920}\end{array}\right]$

  $A4=inv(B)\times inv(A)=\left[\begin{array}{l}\text{ 10}\text{.8369 -30}\text{.8416 32}\text{.4916 22}\text{.5021}\\ \text{ -9}\text{.9010 -0}\text{.1579 3}\text{.0864 9}\text{.2304}\\ \text{ -7}\text{.7213 28}\text{.7573 -27}\text{.5727 -25}\text{.1680}\\ \text{ 5}\text{.7242 6}\text{.0655 -11}\text{.2386 -8}\text{.9117}\end{array}\right]$

Explanation of Solution

Given:The matrix has been given

  $\begin{array}{l}A=rand(4)\\ B=rand(4)\end{array}$

Concept Used:

Given,

  $\begin{array}{l}A=rand(4)\\ B=rand(4)\end{array}$

Calculate the given relationship in MATLAB.

Program:

clc
clear
close all
A=rand(4);
B=rand(4);
A1=inv(A*B)
A2=inv(A)*inv(B)
A3=inv(B*A)
A4=inv(B)*inv(A)
A1-A4
A2-A3

Quarry:

• First, we will define the random matrix A and B.
• Calculate the given relationship.
• Show the difference between two matrices.

To determine

Compute the given relationship using random matrix in MATLAB.

Answer to Problem 1E

The solution is

  $A1=inv((A\times B)\text{'})=\left[\begin{array}{l}\text{ -19}\text{.2699 11}\text{.2507 11}\text{.1342 -15}\text{.2627}\\ \text{ -16}\text{.7260 -16}\text{.1042 0}\text{.8038 21}\text{.3048}\\ \text{ 44}\text{.3982 4}\text{.9491 -19}\text{.0348 3}\text{.2452}\\ \text{ 5}\text{.6507 4}\text{.9232 3}\text{.3463 -13}\text{.5923}\end{array}\right]$

  $A2=inv(A\text{'}\times B\text{'})=\left[\begin{array}{l}\text{ 0}\text{.3349 -59}\text{.7601 0}\text{.3028 19}\text{.2572}\\ \text{ 4}\text{.1133 -20}\text{.7907 1}\text{.7670 0}\text{.5741}\\ \text{ -7}\text{.2267 66}\text{.3396 -8}\text{.7572 -4}\text{.4413}\\ \text{ 4}\text{.1318 62}\text{.3865 14}\text{.1593 -38}\text{.7882}\end{array}\right]$

  $A3=inv(A\text{'})\times inv(B\text{'})=\left[\begin{array}{l}\text{ -19}\text{.2699 11}\text{.2507 11}\text{.1342 -15}\text{.2627}\\ \text{ -16}\text{.7260 -16}\text{.1042 0}\text{.8038 21}\text{.3048}\\ \text{ 44}\text{.3982 4}\text{.9491 -19}\text{.0348 3}\text{.2452}\\ \text{ 5}\text{.6507 4}\text{.9232 3}\text{.3463 -13}\text{.5923}\end{array}\right]$

  $A4=(inv(A)\times inv(B))\text{'}=\left[\begin{array}{l}\text{ 0}\text{.3349 -59}\text{.7601 0}\text{.3028 19}\text{.2572}\\ \text{ 4}\text{.1133 -20}\text{.7907 1}\text{.7670 0}\text{.5741}\\ \text{-7}\text{.2267 66}\text{.3396 -8}\text{.7572 -4}\text{.4413}\\ \text{ 4}\text{.1318 62}\text{.3865 14}\text{.1593 -38}\text{.7882}\end{array}\right]$

Explanation of Solution

Given:The matrix has been given

  $\begin{array}{l}A=rand(4)\\ B=rand(4)\end{array}$

Concept Used:

Given,

  $\begin{array}{l}A=rand(4)\\ B=rand(4)\end{array}$

Calculate the given relationship in MATLAB.

Program:

clc
clear
close all
A=rand(4);
B=rand(4);
A1=inv((A*B)')
A2=inv(A'*B')
A3=inv(A')*inv(B')
A4=(inv(A)*inv(B))'
A1-A4
A2-A3

Quarry:

• First, we will define the random matrix A and B.
• Calculate the given relationship.
• Show the difference between two matrices.