|-1 0 0 0|
|0 -1 3 0|
|0 0 2 0|
|-3 3 -3 2|
Given this matrix A, and the polynom f(x) = x^2 + x + 2, what is the matrix M = f(A) ? Thanks alot!
$\displaystyle \left[\begin{array}{cccc}-1&0&0&0\\0&-1&3&0\\0&0&2&0\\-3&3&-3&2\end{array}\right] $
$\displaystyle f(x)= x^2 + x +2 $
$\displaystyle f(A) = A^2 + A + 2I $ I is the identity
$\displaystyle f(A) =\left[\begin{array}{cccc}-1&0&0&0\\0&-1&3&0\\0&0&2&0\\-3&3&-3&2\end{array}\right]\times \left[\begin{array}{cccc}-1&0&0&0\\0&-1&3&0\\0&0&2&0\\-3&3&-3&2\end{array}\right] $$\displaystyle + \left[\begin{array}{cccc}-1&0&0&0\\0&-1&3&0\\0&0&2&0\\-3&3&-3&2\end{array}\right] + 2 \left[\begin{array}{cccc}1&0&0&0\\0&1&0&0\\0&0&1&0\\0&0& 0&1\end{array}\right]
$