Question

If $A=\left[\begin{array}{ccc}2& -3& -5\\ -1& 4& 5\\ 1& -3& -4\end{array}\right]$, show that A² = A.

Answer 1

$$\begin{gathered} \mathrm{Here}, \\ {A}^2=AA \\ \Rightarrow A^2=\left[\begin{array}{ccc}2& -3& -5\\ -1& 4& 5\\ 1& -3& -4\end{array}\right]\left[\begin{array}{ccc}2& -3& -5\\ -1& 4& 5\\ 1& -3& -4\end{array}\right] \\ \Rightarrow A^2=\left[\begin{array}{ccc}4+3-5& -6-12+15& -10-15+20\\ -2-4+5& 3+16-15& 5+20-20\\ 2+3-4& -3-12+12& -5-15+16\end{array}\right] \\ \Rightarrow A^2=\left[\begin{array}{ccc}2& -3& -5\\ -1& 4& 5\\ 1& -3& -4\end{array}\right] \\ \therefore A^2=A \end{gathered}$$