Question

Let P(x) = x2 - x - 6. Compute P(A) for matrix A = [3, -1and 0 -2]

Answer 1

Given, $P(x)=x^2-x-6$

$A=\begin{bmatrix} 2&-1\\0&-2\end{bmatrix}$

$P(A)=A^2-A-6I$

   $=\begin{bmatrix} 2&-1\\0&-2\end{bmatrix}\begin{bmatrix} 2&-1\\0&-2\end{bmatrix}-\begin{bmatrix} 2&-1\\0&-2\end{bmatrix}-6\begin{bmatrix}1&0\\0&1\end{bmatrix} \\[9pt] = \begin{bmatrix} 9&-1\\0&4\end{bmatrix} +\begin{bmatrix} -3&1\\0&2\end{bmatrix} -\begin{bmatrix} 6&0\\0&6\end{bmatrix} \\[9pt] =\begin{bmatrix} 0&0\\0&0\end{bmatrix}$