Chapter 2, Problem 45E

To determine

To create:

• The variable for the given matrix $\text{A}=\left[\begin{array}{cc}1& 4\\ 3& 2\end{array}\right]$ and perform the operation “3*A”.

• The variable for the given two matrices $\text{A}=\left[\begin{array}{cc}1& 4\\ 3& 2\end{array}\right]$ and $\text{C}=\left[\begin{array}{ccc}3& 2& 5\\ 4& 1& 2\end{array}\right]$ and perform the operation “A*C”.

• To find whether other multiplication operation can be performed in part 2 or not.

Answer to Problem 45E

Solution:

• The variable for the given matrix $\text{A}=\left[\begin{array}{cc}1& 4\\ 3& 2\end{array}\right]$ is “A = [1 4; 3 2]”, and the output of the operation “3*A” is $\left[\begin{array}{cc}3& 12\\ 9& 6\end{array}\right]$.

• The variable for the given two matrices $\text{A}=\left[\begin{array}{cc}1& 4\\ 3& 2\end{array}\right]$ and $\text{C}=\left[\begin{array}{ccc}3& 2& 5\\ 4& 1& 2\end{array}\right]$ are “A = [1 4; 3 2]” and “C = [3 2 5; 4 1 2]”. The output of the operation “A*C” is $\left[\begin{array}{ccc}19& 6& 13\\ 17& 8& 19\end{array}\right]$.

• The variable for the given two matrices $\text{C}=\left[\begin{array}{ccc}3& 2& 5\\ 4& 1& 2\end{array}\right]$ and $\text{B}=\left[\begin{array}{ccc}2& 1& 3\\ 1& 5& 6\\ 3& 6& 0\end{array}\right]$ are “C = [3 2 5; 4 1 2]” and “B = [2 1 3; 1 5 6; 3 6 0]”. The output of the operation “C*B” is $\left[\begin{array}{ccc}23& 43& 21\\ 15& 21& 18\end{array}\right]$.

Explanation of Solution

• The variable for the given matrix $\text{A}=\left[\begin{array}{cc}1& 4\\ 3& 2\end{array}\right]$ and perform the operation “3*A”.

The given matrices are,

$\text{A}=\left[\begin{array}{cc}1& 4\\ 3& 2\end{array}\right]$

$\text{B}=\left[\begin{array}{ccc}2& 1& 3\\ 1& 5& 6\\ 3& 6& 0\end{array}\right]$

And,

$\text{C}=\left[\begin{array}{ccc}3& 2& 5\\ 4& 1& 2\end{array}\right]$

Substitute $\left[\begin{array}{cc}1& 4\\ 3& 2\end{array}\right]$ for A in the expression “3*A”.

$\begin{array}{rll}3*\text{A}& =& 3\left[\begin{array}{cc}1& 4\\ 3& 2\end{array}\right]\\ & =& \left[\begin{array}{cc}1\times 3& 4\times 3\\ 3\times 3& 2\times 3\end{array}\right]\\ & =& \left[\begin{array}{cc}3& 12\\ 9& 6\end{array}\right]\end{array}$

Therefore, the elements of the matrix “3*A” are $\left[\begin{array}{cc}3& 12\\ 9& 6\end{array}\right]$.

Now, verify the expression output through the MATLAB command.

MATLAB Code:

A = [1 4; 3 2]

%Define the command to get the matrix “A”.

3*A

%Define the command to get the output of the expression “3*A”.

Save the MATLAB script with name, chapter2_54793_2_45_1E.m in the current folder. Execute the script by typing the script name at the command window to create the variable for the given matrix $\text{A}=\left[\begin{array}{cc}1& 4\\ 3& 2\end{array}\right]$ and perform the operation “3*A”.

Result:

Therefore, the variable for the given matrix $\text{A}=\left[\begin{array}{cc}1& 4\\ 3& 2\end{array}\right]$ is “A = [1 4; 3 2]”, and the output of the operation “3*A” is $\left[\begin{array}{cc}3& 12\\ 9& 6\end{array}\right]$.

• The variable for the given two matrices $\text{A}=\left[\begin{array}{cc}1& 4\\ 3& 2\end{array}\right]$ and $\text{C}=\left[\begin{array}{ccc}3& 2& 5\\ 4& 1& 2\end{array}\right]$ and perform the operation “A*C”.

The given matrices are,

$\text{A}=\left[\begin{array}{cc}1& 4\\ 3& 2\end{array}\right]$

$\text{B}=\left[\begin{array}{ccc}2& 1& 3\\ 1& 5& 6\\ 3& 6& 0\end{array}\right]$

And,

$\text{C}=\left[\begin{array}{ccc}3& 2& 5\\ 4& 1& 2\end{array}\right]$

Substitute $\left[\begin{array}{cc}1& 4\\ 3& 2\end{array}\right]$ for A and substitute $\left[\begin{array}{ccc}3& 2& 5\\ 4& 1& 2\end{array}\right]$ for C in the expression “A*C”.

$\begin{array}{rll}\text{A*C}& =& \left[\begin{array}{cc}1& 4\\ 3& 2\end{array}\right]\left[\begin{array}{ccc}3& 2& 5\\ 4& 1& 2\end{array}\right]\\ & =& \left[\begin{array}{ccc}\left(1\times 3\right)+\left(4\times 4\right)& \left(1\times 2\right)+\left(4\times 1\right)& \left(1\times 5\right)+\left(4\times 2\right)\\ \left(3\times 3\right)+\left(2\times 4\right)& \left(3\times 2\right)+\left(2\times 1\right)& \left(3\times 5\right)+\left(2\times 2\right)\end{array}\right]\\ & =& \left[\begin{array}{ccc}19& 6& 13\\ 17& 8& 19\end{array}\right]\end{array}$

Therefore, the elements of the matrix “A*C” are $\left[\begin{array}{ccc}19& 6& 13\\ 17& 8& 19\end{array}\right]$.

Now, verify the expression output through the MATLAB command.

MATLAB Code:

A = [1 4; 3 2]

%Define the command to get the matrix “A”.

C = [3 2 5; 4 1 2]

%Define the command to get the matrix “C”.

A*C

%Define the command to get the output of the expression “A*C”.

Save the MATLAB script with name, chapter2_54793_2_45_2E.m in the current folder. Execute the script by typing the script name at the command window to create the variable for the given two matrices $\text{A}=\left[\begin{array}{cc}1& 4\\ 3& 2\end{array}\right]$ and $\text{C}=\left[\begin{array}{ccc}3& 2& 5\\ 4& 1& 2\end{array}\right]$. Perform the operation “A*C”.

Result:

Therefore, the variable for the given two matrices $\text{A}=\left[\begin{array}{cc}1& 4\\ 3& 2\end{array}\right]$ and $\text{C}=\left[\begin{array}{ccc}3& 2& 5\\ 4& 1& 2\end{array}\right]$ are “A = [1 4; 3 2]” and “C = [3 2 5; 4 1 2]”. The output of the operation “A*C” is $\left[\begin{array}{ccc}19& 6& 13\\ 17& 8& 19\end{array}\right]$.

• To find whether other multiplication operation can be performed in part 2 or not.

The given matrices are,

$\text{A}=\left[\begin{array}{cc}1& 4\\ 3& 2\end{array}\right]$

$\text{B}=\left[\begin{array}{ccc}2& 1& 3\\ 1& 5& 6\\ 3& 6& 0\end{array}\right]$

And,

$\text{C}=\left[\begin{array}{ccc}3& 2& 5\\ 4& 1& 2\end{array}\right]$

Substitute $\left[\begin{array}{ccc}2& 1& 3\\ 1& 5& 6\\ 3& 6& 0\end{array}\right]$ for B and substitute $\left[\begin{array}{ccc}3& 2& 5\\ 4& 1& 2\end{array}\right]$ for C in the expression “C*B”.

$\begin{array}{rll}\text{C*B}& =& \left[\begin{array}{ccc}3& 2& 5\\ 4& 1& 2\end{array}\right]\left[\begin{array}{ccc}2& 1& 3\\ 1& 5& 6\\ 3& 6& 0\end{array}\right]\\ & =& \left[\begin{array}{ccc}\left(3\times 2\right)+\left(2\times 1\right)+\left(5\times 3\right)& \left(3\times 1\right)+\left(2\times 5\right)+\left(5\times 6\right)& \left(3\times 3\right)+\left(2\times 6\right)+\left(5\times 0\right)\\ \left(4\times 2\right)+\left(1\times 1\right)+\left(2\times 3\right)& \left(4\times 1\right)+\left(1\times 5\right)+\left(2\times 6\right)& \left(4\times 3\right)+\left(1\times 6\right)+\left(2\times 0\right)\end{array}\right]\\ & =& \left[\begin{array}{ccc}23& 43& 21\\ 15& 21& 18\end{array}\right]\end{array}$

Therefore, the elements of the matrix “C*B” are $\left[\begin{array}{ccc}23& 43& 21\\ 15& 21& 18\end{array}\right]$.

Now, verify the expression output through the MATLAB command.

MATLAB Code:

C = [3 2 5; 4 1 2]

%Define the command to get the matrix “C”.

B = [2 1 3; 1 5 6; 3 6 0]

%Define the command to get the matrix “B”.

C*B

%Define the command to get the output of the expression “C*B”.

Save the MATLAB script with name, chapter2_54793_2_45_3E.m in the current folder. Execute the script by typing the script name at the command window to create the variable for the given two matrices $\text{C}=\left[\begin{array}{ccc}3& 2& 5\\ 4& 1& 2\end{array}\right]$ and $\text{B}=\left[\begin{array}{ccc}2& 1& 3\\ 1& 5& 6\\ 3& 6& 0\end{array}\right]$. The other multiplication operation that can be performed are “C*B”.

Result:

Therefore, the variable for the given two matrices $\text{C}=\left[\begin{array}{ccc}3& 2& 5\\ 4& 1& 2\end{array}\right]$ and $\text{B}=\left[\begin{array}{ccc}2& 1& 3\\ 1& 5& 6\\ 3& 6& 0\end{array}\right]$ are “C = [3 2 5; 4 1 2]” and “B = [2 1 3; 1 5 6; 3 6 0]”. The output of the operation “C*B” is $\left[\begin{array}{ccc}23& 43& 21\\ 15& 21& 18\end{array}\right]$.