Problem Statement
Given $A = \begin{pmatrix} 2 & 0 \\ -1 & 7 \end{pmatrix}$ and $I = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}$, find $A^2 - 9A + 14I$.
Verified Solution & Marking Scheme
Compute A²
$A^2 = \begin{pmatrix} 2 & 0 \\ -1 & 7 \end{pmatrix} \begin{pmatrix} 2 & 0 \\ -1 & 7 \end{pmatrix} = \begin{pmatrix} 4 & 0 \\ -9 & 49 \end{pmatrix}$
Compute 9A and 14I
$9A = \begin{pmatrix} 18 & 0 \\ -9 & 63 \end{pmatrix}, \quad 14I = \begin{pmatrix} 14 & 0 \\ 0 & 14 \end{pmatrix}$
Evaluate Expression
$\begin{pmatrix} 4 - 18 + 14 & 0 \\ -9 - (-9) + 0 & 49 - 63 + 14 \end{pmatrix} = \begin{pmatrix} 0 & 0 \\ 0 & 0 \end{pmatrix} = O$