Question

If $A=\left[\begin{array}{ccc}3& 0& 0\\ 0& 3& 0\\ 0& 0& 3\end{array}\right],$ then A⁴ = _________.

Answer 1

The given matrix is $A=\left[\begin{array}{ccc}3& 0& 0\\ 0& 3& 0\\ 0& 0& 3\end{array}\right].$
Now,

$A^2=A.A=\left[\begin{array}{ccc}3& 0& 0\\ 0& 3& 0\\ 0& 0& 3\end{array}\right]\left[\begin{array}{ccc}3& 0& 0\\ 0& 3& 0\\ 0& 0& 3\end{array}\right]=\left[\begin{array}{ccc}3\times 3+0\times 0+0\times 0& 3\times 0+0\times 3+0\times 0& 3\times 0+0\times 0+0\times 3\\ 0\times 3+3\times 0+0\times 0& 0\times 0+3\times 3+0\times 0& 0\times 0+3\times 0+0\times 3\\ 0\times 3+0\times 0+3\times 0& 0\times 0+0\times 3+3\times 0& 0\times 0+0\times 0+3\times 3\end{array}\right]=\left[\begin{array}{ccc}9& 0& 0\\ 0& 9& 0\\ 0& 0& 9\end{array}\right]$


$\therefore A^4=A^2.A^2=\left[\begin{array}{ccc}9& 0& 0\\ 0& 9& 0\\ 0& 0& 9\end{array}\right]\left[\begin{array}{ccc}9& 0& 0\\ 0& 9& 0\\ 0& 0& 9\end{array}\right]=\left[\begin{array}{ccc}9\times 9+0\times 0+0\times 0& 9\times 0+0\times 9+0\times 0& 9\times 0+0\times 0+0\times 9\\ 0\times 9+9\times 0+0\times 0& 0\times 0+9\times 9+0\times 0& 0\times 0+9\times 0+0\times 9\\ 0\times 9+0\times 0+9\times 0& 0\times 0+0\times 9+9\times 0& 0\times 0+0\times 0+9\times 9\end{array}\right]=\left[\begin{array}{ccc}81& 0& 0\\ 0& 81& 0\\ 0& 0& 81\end{array}\right]$


If $A=\left[\begin{array}{ccc}3& 0& 0\\ 0& 3& 0\\ 0& 0& 3\end{array}\right],$ then A⁴ = $\underline{\left[\begin{array}{ccc}81& 0& 0\\ 0& 81& 0\\ 0& 0& 81\end{array}\right]}$.