#### Provide solution for rd sharma math class 12 chapter Algebra of matrices exercise 4.3 question 65 sub question (iii)

Answer: $B=\left[\begin{array}{cc}0 & b \\ 0 & 0\end{array}\right] , \ \ A=\left[\begin{array}{ll}a & 0 \\ 0 & 0\end{array}\right],$, such that AB=0 but $BA \neq 0$

Hint: Matrix multiplication is only possible, when the number of columns of first matrix is equal to the number of rows of second matrix.

Solution:

$\begin{array}{l} Let \ \ A=\left[\begin{array}{ll} 0 & a \\ 0 & 0 \end{array}\right], B=\left[\begin{array}{ll} b & 0 \\ 0 & 0 \end{array}\right]\\\\ A B=\left[\begin{array}{ll} 0 & a \\ 0 & 0 \end{array}\right]\left[\begin{array}{ll} b & 0 \\ 0 & 0 \end{array}\right]\\\\ =\left[\begin{array}{ll} 0+0 & 0+0 \\ 0+0 & 0+0 \end{array}\right]\\\\ =\left[\begin{array}{ll} 0 & 0 \\ 0 & 0 \end{array}\right]\\\\ A B=0\\\\ \end{array}$

Now consider,

$\begin{array}{l} \\ B A=\left[\begin{array}{ll} b & 0 \\ 0 & 0 \end{array}\right]\left[\begin{array}{ll} 0 & a \\ 0 & 0 \end{array}\right]\\\\ =\left[\begin{array}{cc} 0+0 & a b+0 \\ 0+0 & 0+0 \end{array}\right]\\\\ =\left[\begin{array}{cc} 0 & a b \\ 0 & 0 \end{array}\right]\\\\\\ B A \neq 0 \end{array}$

Hence,

For AB=0 and , we have $BA \neq 0$

$\ \ A=\left[\begin{array}{ll}a & 0 \\ 0 & 0\end{array}\right], B=\left[\begin{array}{cc}0 & b \\ 0 & 0\end{array}\right]$