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

Answers (1)

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]

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