Question

If matrix $A=[a_{ij}{]}_{2\times 2}$, where $a_{ij}=\{\begin{array}{ll}1& \text{if i=j }\\ 0& \text{if i=j }\end{array}$ then $A^2$ is equal to

• A. $I$
• B. $A$
• C. $O$
• D. none of these

Answer 1

Answer: (A)

$a_{11}=0,a_{12}=1,a_{21}=1,a_{22}=0$

$\therefore A=\left[\begin{array}{cc}0& 1\\ 1& 0\end{array}\right]$

$\therefore A^2=\left[\begin{array}{cc}0& 1\\ 1& 0\end{array}\right]\left[\begin{array}{cc}0& 1\\ 1& 0\end{array}\right]=\left[\begin{array}{cc}0+1& 0+0\\ 0+0& 1+0\end{array}\right]=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]=I$