#### Please solve RD Sharma class 12 chapter Algebra of matrices exercise 4.3 question 65 sub question (ii) maths textbook solution

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

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:

Let

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

Hence, AB=0

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