Chapter 2, Problem 2.5P

To determine

To find:

The value of the expressions A.*B, A*B and B*A by hand and then verify the results in MATLAB.

Answer to Problem 2.5P

Solution:

The values of the expressions A.*B, A*B, and B*A are $\left[\begin{array}{cc}1& 8\\ -3& 0\end{array}\right]$, $\left[\begin{array}{cc}-3& 2\\ 0& 6\end{array}\right]$, and $\left[\begin{array}{cc}7& 10\\ -1& -4\end{array}\right]$ respectively.

Explanation of Solution

The given matrices are,

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

And,

$\text{B}=\left[\begin{array}{cc}1& 2\\ -1& 0\end{array}\right]$

Substitute $\left[\begin{array}{cc}1& 4\\ 3& 3\end{array}\right]$ for A and $\left[\begin{array}{cc}1& 2\\ -1& 0\end{array}\right]$ for B in the expression “A.*B”.

$\begin{array}{lll}\text{A}\text{.*B}& =& \left[\begin{array}{cc}1& 4\\ 3& 3\end{array}\right].*\left[\begin{array}{cc}1& 2\\ -1& 0\end{array}\right]\\ & =& \left[\begin{array}{cc}1\times 1& 4\times 2\\ 3\times -1& 3\times 0\end{array}\right]\\ & =& \left[\begin{array}{cc}1& 8\\ -3& 0\end{array}\right]\end{array}$

Substitute $\left[\begin{array}{cc}1& 4\\ 3& 3\end{array}\right]$ for A and $\left[\begin{array}{cc}1& 2\\ -1& 0\end{array}\right]$ for B in the expression “A*B”.

$\begin{array}{rll}\text{A*B}& =& \left[\begin{array}{cc}1& 4\\ 3& 3\end{array}\right]*\left[\begin{array}{cc}1& 2\\ -1& 0\end{array}\right]\\ & =& \left[\begin{array}{cc}\left(1\times 1\right)+\left(4\times -1\right)& \left(1\times 2\right)+\left(4\times 0\right)\\ \left(3\times 1\right)+\left(3\times -1\right)& \left(3\times 2\right)+\left(3\times 0\right)\end{array}\right]\\ & =& \left[\begin{array}{cc}-3& 2\\ 0& 6\end{array}\right]\end{array}$

Similarly substitute $\left[\begin{array}{cc}1& 4\\ 3& 3\end{array}\right]$ for A and $\left[\begin{array}{cc}1& 2\\ -1& 0\end{array}\right]$ for B in the expression “B*A”.

$\begin{array}{rll}\text{B*A}& =& \left[\begin{array}{cc}1& 2\\ -1& 0\end{array}\right]*\left[\begin{array}{cc}1& 4\\ 3& 3\end{array}\right]\\ & =& \left[\begin{array}{cc}\left(1\times 1\right)+\left(2\times 3\right)& \left(1\times 4\right)+\left(2\times 3\right)\\ \left(-1\times 1\right)+\left(0\times 3\right)& \left(-1\times 4\right)+\left(0\times 3\right)\end{array}\right]\\ & =& \left[\begin{array}{cc}7& 10\\ -1& -4\end{array}\right]\end{array}$

Now, verifying the expression output by using MATLAB command.

MATLAB Code:

A = [1 4;3 3]

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

B = [1 2;-1 0]

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

A.*B

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

A*B

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

B*A

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

Save the MATLAB script with name, chapter2_45251_2_5P.m in the current folder. Execute the program by typing the script name at the command window to find the value of the expressions A.*B, A*B and B*A by hand and then verify the results in MATLAB.

Result:

Therefore, the values of the expressions A.*B, A*B, and B*A are $\left[\begin{array}{cc}1& 8\\ -3& 0\end{array}\right]$, $\left[\begin{array}{cc}-3& 2\\ 0& 6\end{array}\right]$, and $\left[\begin{array}{cc}7& 10\\ -1& -4\end{array}\right]$ respectively.