Question

If $A=\left[\begin{array}{ccc}2& 0& 1\\ 2& 1& 3\\ 1& -1& 0\end{array}\right]$, then find the value of $A^2-5A+6I$

Answer 1

$A=\left[\begin{array}{ccc}2& 0& 1\\ 2& 1& 3\\ 1& -1& 0\end{array}\right]$

$A^2=\left[\begin{array}{ccc}2& 0& 1\\ 2& 1& 3\\ 1& -1& 0\end{array}\right]\times \left[\begin{array}{ccc}2& 0& 1\\ 2& 1& 3\\ 1& -1& 0\end{array}\right]$

$\Rightarrow A^2=\left[\begin{array}{ccc}5& -1& 2\\ 9& -2& 5\\ 0& -1& -2\end{array}\right]$

The to find the value of $A^2-5A+6I$

$=\left[\begin{array}{ccc}5& -1& 2\\ 9& -2& 5\\ 0& -1& -2\end{array}\right]-5\left[\begin{array}{ccc}2& 0& 1\\ 2& 1& 3\\ 1& -1& 0\end{array}\right]+6\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]$

$=\left[\begin{array}{ccc}-1& -1& -3\\ -1& -1& -10\\ -5& 4& 4\end{array}\right]$