Question

If $A^2+2A+10=0$
$A=\left[\begin{array}{ccc}1& 12& 15\\ 2& 10& 4\\ 3& 9& 5\end{array}\right]$ find $A^{-1}$

Answer 1

$A^2+2A+10=0$

$A+AA+2\left(A^{-1}A\right)+10A^{-1}=\left(0\right)A^{-1}$

$A+2I+10A^{-1}=0$

$A^{-1}=\frac{-1}{10}(A+2I)$

$=\frac{-1}{10}$$\left|\begin{array}{ccc}1& 12& 15\\ 2& 10& 4\\ 3& 9& 5\end{array}\right|+\left|\begin{array}{ccc}2& 0& 0\\ 0& 2& 0\\ 0& 0& 2\end{array}\right|$

$A^{-1}=\frac{-1}{10}\left|\begin{array}{ccc}3& 12& 15\\ 2& 12& 4\\ 3& 9& 7\end{array}\right|$