Question

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

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

Answer 1

Answer: (A)

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

$\therefore A=\begin{bmatrix}0&1\\ 1&0\end{bmatrix}$

$\therefore A^{2}=\begin{bmatrix}0&1\\ 1&0\end{bmatrix}\begin{bmatrix}0&1\\ 1&0\end{bmatrix}=\begin{bmatrix}0+1&0+0\\ 0+0&1+0\end{bmatrix}=\begin{bmatrix}1&0\\ 0&1\end{bmatrix}=I$