Question

If $A=\left[\begin{array}{ccc}2& 0& 1\\ 2& 1& 3\\ 1& -1& 0\end{array}\right]$, find $A^2-5A+4I$ and hence find a matrix $X$ such that $A^2-5A+4I+X=0$.

Answer 1

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

$A^2-5A+4I=\left[\begin{array}{ccc}5& -1& 2\\ 9& -2& 5\\ 0& -1& -2\end{array}\right]-5\left[\begin{array}{ccc}2& 0& 1\\ 2& 1& 3\\ 1& -1& 0\end{array}\right]+4\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]=\left[\begin{array}{ccc}-1& -1& -3\\ -1& -3& -10\\ -5& 4& 2\end{array}\right]$.

therefore,

$X=\left[\begin{array}{ccc}1& 1& 3\\ 1& 3& 10\\ 5& -4& -2\end{array}\right]$