Question

Find $A^2-5A+611$ if $A=\left[\begin{array}{ccc}2& 0& 1\\ 2& 1& 3\\ 1& -1& 0\end{array}\right]$

Answer 1

Let the problem,

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

Now,

we have to calculate,

$A^2-5A+6I$ ------ (i)

For this $A^2=\left[\begin{array}{ccc}2& 0& 1\\ 2& 1& 3\\ 1& -1& 0\end{array}\right]\left[\begin{array}{ccc}2& 0& 1\\ 2& 1& 3\\ 1& -1& 0\end{array}\right]=\left[\begin{array}{ccc}5& 1& 2\\ 9& -2& 5\\ 0& -1& -2\end{array}\right]$

Now, from (i)

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

$==\left[\begin{array}{ccc}5& 1& 2\\ 9& -2& 5\\ 0& -1& -2\end{array}\right]-\left[\begin{array}{ccc}10& 0& 5\\ 10& 5& 15\\ 5& -5& 0\end{array}\right]+\left[\begin{array}{ccc}6& 0& 0\\ 0& 6& 0\\ 0& 0& 6\end{array}\right]$

$=\left[\begin{array}{ccc}11& 1& 2\\ 9& 4& 5\\ 0& -1& 4\end{array}\right]-\left[\begin{array}{ccc}10& 0& 5\\ 10& 5& 15\\ 5& -5& 0\end{array}\right]$

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

Hence $A^2-5A+6I=\left[\begin{array}{ccc}1& 1& -3\\ -1& -1& -10\\ -5& 4& 4\end{array}\right]$