Question
If \[ A = \begin{bmatrix} 1 & 0 \\ -1 & 7 \end{bmatrix} \] find \(k\) such that \[ A^2 – 8A + kI = O. \]
Solution
Step 1: Compute \(A^2\)
\[ A^2 = \begin{bmatrix} 1 & 0 \\ -1 & 7 \end{bmatrix} \begin{bmatrix} 1 & 0 \\ -1 & 7 \end{bmatrix} = \begin{bmatrix} 1\cdot1 + 0(-1) & 1\cdot0 + 0\cdot7 \\ (-1)\cdot1 + 7(-1) & (-1)\cdot0 + 7\cdot7 \end{bmatrix} = \begin{bmatrix} 1 & 0 \\ -8 & 49 \end{bmatrix} \]Step 2: Form Expression
\[ A^2 – 8A + kI = \begin{bmatrix} 1 & 0 \\ -8 & 49 \end{bmatrix} – \begin{bmatrix} 8 & 0 \\ -8 & 56 \end{bmatrix} + \begin{bmatrix} k & 0 \\ 0 & k \end{bmatrix} \]Step 3: Simplify
\[ = \begin{bmatrix} 1 – 8 + k & 0 \\ -8 + 8 & 49 – 56 + k \end{bmatrix} = \begin{bmatrix} k – 7 & 0 \\ 0 & k – 7 \end{bmatrix} \]Step 4: Compare with Zero Matrix
\[ k – 7 = 0 \Rightarrow k = 7 \]Final Answer
\[
k = 7
\]