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

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

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

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

$\begin{array}{l} A B=\left[\begin{array}{ll} a & 0 \\ 0 & 0 \end{array}\right]\left[\begin{array}{ll} 0 & b \\ 0 & 0 \end{array}\right]\\\\ =\left[\begin{array}{cc} 0+0 & a b+0 \\ 0+0 & 0+0 \end{array}\right]\\\\ A B=\left[\begin{array}{cc} 0 & a b \\ 0 & 0 \end{array}\right] ... (i)\\\\ B A=\left[\begin{array}{ll} 0 & b \\ 0 & 0 \end{array}\right]\left[\begin{array}{ll} a & 0 \\ 0 & 0 \end{array}\right]\\\\ =\left[\begin{array}{ll} 0+0 & 0+0 \\ 0+0 & 0+0 \end{array}\right]\\\\ B A=\left[\begin{array}{ll} 0 & 0 \\ 0 & 0 \end{array}\right] \quad \cdots (ii) \ \end{array}$

From equation i & ii

$AB \neq BA$

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