#### Explain Solution R. D. Sharma Class 12 Chapter 4 Algebra of Matrices Exercise Multiple Choice Questions Question 40 Maths Textbook Solution.

Answer: $I$

Given: If matrix $A=\left[a_{i j}\right]_{2 \times 2} \text { where } a_{i j}=\left\{\begin{array}{ll} 1 & \text { if } i \neq j \\ 0 & \text { if } i=j \end{array} \text { then } A^{2}\right.$

Hint: find $A^{2}$

Solution:

We have $A=\left[a_{i j}\right]_{2 \times 2} \text { where } \begin{cases}1 & \text { if } i \neq j \\ 0 & \text { if } i=j\end{cases}$

Let A be a square matrix

$\text { i.e. } A=\left[\begin{array}{ll} a_{11} & a_{12} \\ a_{21} & a_{22} \end{array}\right]$

Now for $i \neq j \text { i.e. } a_{12}=1 \text { and } a_{21}=1$

And for $i=j \text { i.e. } a_{11}=0 \text { and } a_{22}=0$

\begin{aligned} &\therefore A=\left[\begin{array}{ll} 0 & 1 \\ 1 & 0 \end{array}\right] \\ &\therefore A^{2}=\left[\begin{array}{ll} 0 & 1 \\ 1 & 0 \end{array}\right]\left[\begin{array}{ll} 0 & 1 \\ 1 & 0 \end{array}\right]=\left[\begin{array}{ll} 1 & 0 \\ 0 & 1 \end{array}\right]=I \end{aligned}