Question
If \[ A = \begin{bmatrix} 3 & 2 & 0 \\ 1 & 4 & 0 \\ 0 & 0 & 5 \end{bmatrix} \] show that \[ A^2 – 7A + 10I_3 = O. \]
Solution
Step 1: Compute \(A^2\)
\[ A^2 = \begin{bmatrix} 11 & 14 & 0 \\ 7 & 18 & 0 \\ 0 & 0 & 25 \end{bmatrix} \]Step 2: Form Expression
\[ A^2 – 7A + 10I_3 = \begin{bmatrix} 11 & 14 & 0 \\ 7 & 18 & 0 \\ 0 & 0 & 25 \end{bmatrix} – \begin{bmatrix} 21 & 14 & 0 \\ 7 & 28 & 0 \\ 0 & 0 & 35 \end{bmatrix} + \begin{bmatrix} 10 & 0 & 0 \\ 0 & 10 & 0 \\ 0 & 0 & 10 \end{bmatrix} \]Step 3: Simplify
\[ = \begin{bmatrix} 11 – 21 + 10 & 14 – 14 + 0 & 0 \\ 7 – 7 + 0 & 18 – 28 + 10 & 0 \\ 0 & 0 & 25 – 35 + 10 \end{bmatrix} \] \[ = \begin{bmatrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{bmatrix} \]Final Result
\[
A^2 – 7A + 10I_3 = O
\]
Hence proved.