# Solve this following

Question:

Show that $A B \neq B A$ in each of the following cases:

$A=\left[\begin{array}{cc}5 & -1 \\ 6 & 7\end{array}\right]$ and $B=\left[\begin{array}{ll}2 & 1 \\ 3 & 4\end{array}\right]$

Solution:

Given : $A=\left[\begin{array}{cc}5 & -1 \\ 6 & 7\end{array}\right]$ and $B=\left[\begin{array}{cc}2 & 1 \\ 3 & 4\end{array}\right]$

Matrix A is of order $2 \times 2$ and Matrix B is of order $2 \times 2$

To show : matrix $\mathrm{AB} \neq \mathrm{BA}$

Formula used:

Where $c_{i j}=a_{i 1} b_{1 j}+a_{i 2} b_{2 j}+a_{i 3} b_{3 j}+\ldots \ldots \ldots \ldots \ldots . . a_{i n} b_{n j}$

If $\mathrm{A}$ is a matrix of order $\mathrm{a} \times \mathrm{b}$ and $\mathrm{B}$ is a matrix of order $\mathrm{c}_{\times} \mathrm{d}$, then matrix $\mathrm{AB}$ exists and is of order $a \times d$, if and only if $b=$ $c$

If $A$ is a matrix of order $a \times b$ and $B$ is a matrix of order $c \times d$, then matrix $B A$ exists and is of order $c \times b$, if and only if $d=$ a

For matrix $A B, a=2, b=c=2, d=2$, thus matrix $A B$ is of order $2 \times 2$

Matrix $\mathrm{AB}=\left[\begin{array}{cc}5 & -1 \\ 6 & 7\end{array}\right] \times\left[\begin{array}{ll}2 & 1 \\ 3 & 4\end{array}\right]=\left[\begin{array}{cc}5(2)-1(3) & 5(1)-1(4) \\ 6(2)+7(3) & 6(1)+7(4)\end{array}\right]$

Matrix $\mathrm{AB}=\left[\begin{array}{cc}10-3 & 5-4 \\ 12+21 & 6+28\end{array}\right]=\left[\begin{array}{cc}7 & 1 \\ 33 & 34\end{array}\right]$

Matrix $\mathrm{AB}=\left[\begin{array}{cc}7 & 1 \\ 33 & 34\end{array}\right]$

For matrix $\mathrm{BA}, \mathrm{a}=2, \mathrm{~b}=\mathrm{c}=2, \mathrm{~d}=2$, thus matrix $\mathrm{BA}$ is of order $2 \times 2$

Matrix $\mathrm{BA}=\left[\begin{array}{ll}2 & 1 \\ 3 & 4\end{array}\right] \times\left[\begin{array}{cc}5 & -1 \\ 6 & 7\end{array}\right]=\left[\begin{array}{ll}2(5)+1(6) & 2(-1)+1(7) \\ 3(5)+4(6) & 3(-1)+4(7)\end{array}\right]$ Matrix BA $=\left[\begin{array}{cc}10+6 & -2+7 \\ 15+24 & -3+28\end{array}\right]=\left[\begin{array}{cc}16 & 5 \\ 39 & 25\end{array}\right]$

Matrix BA $=\left[\begin{array}{cc}16 & 5 \\ 39 & 25\end{array}\right]$

Matrix $B A=\left[\begin{array}{cc}16 & 5 \\ 39 & 25\end{array}\right]$ and Matrix $A B=\left[\begin{array}{cc}7 & 1 \\ 33 & 34\end{array}\right]$

Matrix $\mathrm{AB} \neq \mathrm{BA}$