Question

If $A=\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ a& b& -1\end{array}\right]$, then A² is equal to
(a) a null matrix
(b) a unit matrix
(c) −A
(d) A

Answer 1

$\left(b\right)$ a unit matrix


$$\begin{gathered} A^2=AA \\ \Rightarrow A^2=\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ a& b& -1\end{array}\right]\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ a& b& -1\end{array}\right] \\ \Rightarrow A^2=\left[\begin{array}{ccc}1+0+0& 0+0+0& 0+0-0\\ 0+0+0& 0+1+0& 0+0-0\\ a+0-a& 0+b-b& 0+0+1\end{array}\right] \\ \Rightarrow A^2=\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right] \end{gathered}$$