Answer to Question #190362 in Linear Algebra for Regomoditswe Dibob

Question

Let P(x) = x 2 − x − 6. Compute P(A) for A =  3 −1 0 −2 

Answer 1

Given,

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

$=x^2-3x+2x-6$

$=x(x-3)+2(x-3)$

$=(x-3)(x+2)$

As here matrix $A=\begin{bmatrix} 3 & -1\\ 0 &2 \end{bmatrix}$

Matrix order is $2\times 2$

Hence, $A^2=\begin{bmatrix} 3 & -1\\ 0 & 2 \end{bmatrix}\begin{bmatrix} 3 & -1\\ 0 & 2 \end{bmatrix}$

$=\begin{bmatrix} 9 & (-3-2)\\ 0 & 4 \end{bmatrix}$

$=\begin{bmatrix} 9 & (-5)\\ 0 & 4 \end{bmatrix}$

Now, substituting the values,

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

Hence,

$=\begin{bmatrix} 9 & (-5)\\ 0 & 4 \end{bmatrix}-\begin{bmatrix} 3 & -1\\ 0 &2 \end{bmatrix}-6\begin{bmatrix} 1 & 0\\ 0 &1 \end{bmatrix}$

$=\begin{bmatrix} 0 & -4\\ 0 &0 \end{bmatrix}$

Hence P(A)$=\begin{bmatrix} 0 & -4\\ 0 &0 \end{bmatrix}$ .