Question

If matrix $A={\left[a_{ij}\right]}_{2\times 2}$, where $a_{ij}=\left\{\begin{array}{ll}1& ,\mathrm{if}i\ne j\\ 0& ,\mathrm{if}i=j\end{array}\right.$, then A² is equal to

(a) I
(b) A
(c) O
(d) −I

Answer 1

Given: $a_{ij}=\left\{\begin{array}{ll}1& ,\mathrm{if}i\ne j\\ 0& ,\mathrm{if}i=j\end{array}\right.$

$$\begin{gathered} \therefore a_{11}=0 \\ {a}_{12}=1 \\ {a}_{21}=1 \\ {a}_{22}=0 \\ \mathrm{Therefore},\mathrm{matrix}A=\left[\begin{array}{cc}0& 1\\ 1& 0\end{array}\right] \\ {A}^2=\left[\begin{array}{cc}0& 1\\ 1& 0\end{array}\right]\left[\begin{array}{cc}0& 1\\ 1& 0\end{array}\right] \\ =\left[\begin{array}{cc}0+1& 0+0\\ 0+0& 1+0\end{array}\right] \\ =\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right] \\ =I \end{gathered}$$

Hence, the correct option is (a).