Answer
$f(A)=A^{3} -2A^{2}+5A-10I=\left[\begin{array}{ccc}
4&6&-12\\
3&-11&21\\
-15&9&25
\end{array}\right]$
Work Step by Step
Given $f(x)=x^{3} -2x^{2}+5x-10$ and the matrix
$A=\left[\begin{array}{ccc}
2&1&-1\\
1&0&2\\
-1&1&3
\end{array}\right]$
We have
$A^{2}=\left[\begin{array}{ccc}
6&1&-3\\
0&3&5\\
-4&2&12
\end{array}\right]$
and $A^{3}=\left[\begin{array}{ccc}
16&3&-13\\
-2&5&21\\
-18&8&44
\end{array}\right]$
Thus we have for the matrix $A$,
$f(A)=A^{3} -2A^{2}+5A-10I=\left[\begin{array}{ccc}
4&6&-12\\
3&-11&21\\
-15&9&25
\end{array}\right]$
