Chapter 2, Problem 1E

Generate random $5\times 5$ matrices with integer entries by setting

$A=\text{round}\left(10*\text{rand}\left(5\right)\right)$

and

$B=\text{round}\left(20*\text{rand}\left(5\right)\right)-10$

Use MATLAB to compute each of the pairs of numbers that follow. In each case, check whether in first number is equal to the second.

(a) $\det \left(A\right)\det \left(A^T\right)$

(b) $\det \left(A+B\right)\det \left(A\right)+\det \left(B\right)$

(c) $\det \left(AB\right)\det \left(A\right)+\det \left(B\right)$

(d) $\det \left(A^T{B}^T\right)\det \left(A^T\right)\det \left(B^T\right)$

(e) $\det \left(A^{-1}\right)1/\det \left(A\right)$

(f) $\det \left(A{B}^{-1}\right)\det \left(A\right)/\det \left(B\right)$

To determine

Calculate the given relationship.

Answer to Problem 1E

The solution of the system is

  $\begin{array}{l}\det (A)=\text{-54164}\\ \det (A^T)=\text{-54164}\end{array}$

Explanation of Solution

Given:The matrix has been given

  $\begin{array}{l}A=round(10\times rand(5))\\ B=round(20\times rand(5))-10\end{array}$

Concept Used:

Given,

  $\begin{array}{l}A=round(10\times rand(5))\\ B=round(20\times rand(5))-10\end{array}$

Using given information calculate the determinant of the matrix.

  $\begin{array}{l}\det (A)=\text{-54164}\\ \det (A^T)=\text{-54164}\end{array}$

Program:

clc
clear
close all
A=round(10*rand(6));
B=round(20*rand(6))-10;
a=det(A);
b=det(A');

Quarry:

• First, we have defined the given matrix A and B.
• Then Calculate the determinant of the matrices.

To determine

Calculate the given relationship.

Answer to Problem 1E

The solution of the system is

  $\begin{array}{l}\text{det(A+B)= -1}\text{.0914}\times {\text{10}}^{\text{-11}}\\ \text{det(A)+det(B)=77653}0\end{array}$

Explanation of Solution

Given:The matrix has been given

  $\begin{array}{l}A=round(10\times rand(5))\\ B=round(20\times rand(5))-10\end{array}$

Concept Used:

Given,

  $\begin{array}{l}A=round(10\times rand(5))\\ B=round(20\times rand(5))-10\end{array}$

Using given information calculate the determinant of the matrix.

  $\begin{array}{l}\text{det(A+B)= -1}\text{.0914}\times {\text{10}}^{\text{-11}}\\ \text{det(A)+det(B)=77653}0\end{array}$

Program:

clc
clear
close all
A=round(10*rand(6));
B=round(20*rand(6))-10;
a=det(A);
b=det(A');
c=det(A)-det(A');
d=det(A-B);
e=det(A)-det(B);
f=det(A-B)-(det(A)-det(B));
g=det(A*B);
h=det(A)*det(B);
i=det(A*B)-det(A)*det(B);
j=det(A'*B);
k=det(A')*det(B);
l=det(A'*B)-det(A')*det(B);
m=det(A^-1);
n=1/det(A);
o=det(A^-1)-1/det(A);
p=det(A*B^-1);
q=det(A)/det(B);
r=det(A*B^-1)-det(A)/det(B);

Quarry:

• First, we have defined the given matrix A and B.
• Then Calculate the determinant of the matrices.

To determine

Calculate the given relationship.

Answer to Problem 1E

The solution of the system is

  $\begin{array}{l}\det (AB)=\text{112814}\\ \det (A)\det (B)=\text{66372}0\end{array}$

Explanation of Solution

Given:The matrix has been given

  $\begin{array}{l}A=round(10\times rand(5))\\ B=round(20\times rand(5))-10\end{array}$

Concept Used:

Given,

  $\begin{array}{l}A=round(10\times rand(5))\\ B=round(20\times rand(5))-10\end{array}$

Using given information calculate the determinant of the matrix.

  $\begin{array}{l}\det (AB)=\text{112814}\\ \det (A)\det (B)=\text{66372}0\end{array}$

Program:

clc
clear
close all
A=round(10*rand(6));
B=round(20*rand(6))-10;
a=det(A);
b=det(A');
c=det(A)-det(A');
d=det(A-B);
e=det(A)-det(B);
f=det(A-B)-(det(A)-det(B));
g=det(A*B);
h=det(A)*det(B);
i=det(A*B)-det(A)*det(B);
j=det(A'*B);
k=det(A')*det(B);
l=det(A'*B)-det(A')*det(B);
m=det(A^-1);
n=1/det(A);
o=det(A^-1)-1/det(A);
p=det(A*B^-1);
q=det(A)/det(B);
r=det(A*B^-1)-det(A)/det(B);

Quarry:

• First, we have defined the given matrix A and B.
• Then Calculate the determinant of the matrices.

To determine

Calculate the given relationship.

Answer to Problem 1E

The solution of the system is

  $\begin{array}{l}\det (A^T{B}^T)=\text{-2}\text{.2792}\times {\text{10}}^{\text{9}}\\ \det (A^T)\det (B^T)=\text{-2}\text{.2792}\times {\text{10}}^{\text{9}}\end{array}$

Explanation of Solution

Given:The matrix has been given

  $\begin{array}{l}A=round(10\times rand(5))\\ B=round(20\times rand(5))-10\end{array}$

Concept Used:

Given,

  $\begin{array}{l}A=round(10\times rand(5))\\ B=round(20\times rand(5))-10\end{array}$

Using given information calculate the determinant of the matrix.

  $\begin{array}{l}\det (A^T{B}^T)=\text{-2}\text{.2792}\times {\text{10}}^{\text{9}}\\ \det (A^T)\det (B^T)=\text{-2}\text{.2792}\times {\text{10}}^{\text{9}}\end{array}$

Program:

clc
clear
close all
A=round(10*rand(6));
B=round(20*rand(6))-10;
a=det(A);
b=det(A');
c=det(A)-det(A');
d=det(A-B);
e=det(A)-det(B);
f=det(A-B)-(det(A)-det(B));
g=det(A*B);
h=det(A)*det(B);
i=det(A*B)-det(A)*det(B);
j=det(A'*B);
k=det(A')*det(B);
l=det(A'*B)-det(A')*det(B);
m=det(A^-1);
n=1/det(A);
o=det(A^-1)-1/det(A);
p=det(A*B^-1);
q=det(A)/det(B);
r=det(A*B^-1)-det(A)/det(B);

Quarry:

• First, we have defined the given matrix A and B.
• Then Calculate the determinant of the matrices.

To determine

Calculate the given relationship.

Answer to Problem 1E

The solution of the system is

  $\begin{array}{l}\det (A^{-1})=\text{-4}\text{.7684}\times {\text{10}}^{-6}\\ 1/\det (A)=\text{-2}\text{.2792}\times {\text{10}}^{\text{9}}\end{array}$

Explanation of Solution

Given:The matrix has been given

  $\begin{array}{l}A=round(10\times rand(5))\\ B=round(20\times rand(5))-10\end{array}$

Concept Used:

Given,

  $\begin{array}{l}A=round(10\times rand(5))\\ B=round(20\times rand(5))-10\end{array}$

Using given information calculate the determinant of the matrix.

  $\begin{array}{l}\det (A^{-1})=\text{-4}\text{.7684}\times {\text{10}}^{-6}\\ 1/\det (A)=\text{-2}\text{.2792}\times {\text{10}}^{\text{9}}\end{array}$

Program:

clc
clear
close all
A=round(10*rand(6));
B=round(20*rand(6))-10;
a=det(A);
b=det(A');
c=det(A)-det(A');
d=det(A-B);
e=det(A)-det(B);
f=det(A-B)-(det(A)-det(B));
g=det(A*B);
h=det(A)*det(B);
i=det(A*B)-det(A)*det(B);
j=det(A'*B);
k=det(A')*det(B);
l=det(A'*B)-det(A')*det(B);
m=det(A^-1);
n=1/det(A);
o=det(A^-1)-1/det(A);
p=det(A*B^-1);
q=det(A)/det(B);
r=det(A*B^-1)-det(A)/det(B);

Quarry:

• First, we have defined the given matrix A and B.
• Then Calculate the determinant of the matrices.

To determine

Calculate the given relationship.

Answer to Problem 1E

The solution of the system is

  $\begin{array}{l}\det (A{B}^{-1})=\text{-2}\text{.2792}\times {\text{10}}^{\text{9}}\\ \det (A)/\det (B)=\text{-8}\text{.1062}\times {\text{10}}^{-6}\end{array}$

Explanation of Solution

Given:The matrix has been given

  $\begin{array}{l}A=round(10\times rand(5))\\ B=round(20\times rand(5))-10\end{array}$

Concept Used:

Given,

  $\begin{array}{l}A=round(10\times rand(5))\\ B=round(20\times rand(5))-10\end{array}$

Using given information calculate the determinant of the matrix.

  $\begin{array}{l}\det (A{B}^{-1})=\text{-2}\text{.2792}\times {\text{10}}^{\text{9}}\\ \det (A)/\det (B)=\text{-8}\text{.1062}\times {\text{10}}^{-6}\end{array}$

Program:

clc
clear
close all
A=round(10*rand(6));
B=round(20*rand(6))-10;
a=det(A);
b=det(A');
c=det(A)-det(A');
d=det(A-B);
e=det(A)-det(B);
f=det(A-B)-(det(A)-det(B));
g=det(A*B);
h=det(A)*det(B);
i=det(A*B)-det(A)*det(B);
j=det(A'*B);
k=det(A')*det(B);
l=det(A'*B)-det(A')*det(B);
m=det(A^-1);
n=1/det(A);
o=det(A^-1)-1/det(A);
p=det(A*B^-1);
q=det(A)/det(B);
r=det(A*B^-1)-det(A)/det(B);

Quarry:

• First, we have defined the given matrix A and B.
• Then Calculate the determinant of the matrices.