Question:
If $A=\left[\begin{array}{rrr}1 & 0 & 0 \\ 0 & 1 & 0 \\ a & b & -1\end{array}\right]$, then $A^{2}$ is equal to
(a) a : matrix
(b) a unit matrix
(c) $-A$
(d) $A$
Solution:
(b) a unit matrix
$A^{2}=A A$
$\Rightarrow A^{2}=\left[\begin{array}{ccc}1 & 0 & 0 \\ 0 & 1 & 0 \\ a & b & -1\end{array}\right]\left[\begin{array}{ccc}1 & 0 & 0 \\ 0 & 1 & 0 \\ a & b & -1\end{array}\right]$
$\Rightarrow A^{2}=\left[\begin{array}{lll}1+0+0 & 0+0+0 & 0+0-0 \\ 0+0+0 & 0+1+0 & 0+0-0 \\ a+0-a & 0+b-b & 0+0+1\end{array}\right]$
$\Rightarrow A^{2}=\left[\begin{array}{lll}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{array}\right]$
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