Question

If $f\left(x\right)=x^2-5x+6$, and
$A=\left[\begin{array}{ccc}2& 0& 1\\ 2& 1& 3\\ 1& -1& 0\end{array}\right]$
$f\left(A\right)=\left[\begin{array}{ccc}1& -1& a\\ -1& -1& b\\ -5& 4& 4\end{array}\right]$
find $a-b?$

Answer 1

Given $f\left(x\right)=x^2-5x+6$, and
$A=\left[\begin{array}{ccc}2& 0& 1\\ 2& 1& 3\\ 1& -1& 0\end{array}\right]$
$f\left(A\right)=\left[\begin{array}{ccc}1& -1& a\\ -1& -1& b\\ -5& 4& 4\end{array}\right]$
$\Rightarrow f\left(A\right)=A^2-5A+6I$
$\Rightarrow \left[\begin{array}{ccc}1& -1& a\\ -1& -1& b\\ -5& 4& 4\end{array}\right]=\left[\begin{array}{ccc}5& -1& 2\\ 9& -2& 5\\ 0& -1& -2\end{array}\right]-\left[\begin{array}{ccc}10& 0& 5\\ 10& 5& 15\\ 5& -5& 0\end{array}\right]+\left[\begin{array}{ccc}6& 0& 0\\ 0& 6& 0\\ 0& 0& 6\end{array}\right]$
$\Rightarrow \left[\begin{array}{ccc}1& -1& a\\ -1& -1& b\\ -5& 4& 4\end{array}\right]=\left[\begin{array}{ccc}1& -1& -3\\ -1& -1& -10\\ -5& 4& 4\end{array}\right]$
$\Rightarrow a=-3$ and $b=-10$
$\therefore a-b=7$