Question
Show that the matrix \[ A = \begin{bmatrix} 5 & 3 \\ 12 & 7 \end{bmatrix} \] is a root of the equation \[ A^2 – 12A – I = O. \]
Solution
Step 1: Compute \(A^2\)
\[ A^2 = \begin{bmatrix} 5 & 3 \\ 12 & 7 \end{bmatrix} \begin{bmatrix} 5 & 3 \\ 12 & 7 \end{bmatrix} = \begin{bmatrix} 5\cdot5 + 3\cdot12 & 5\cdot3 + 3\cdot7 \\ 12\cdot5 + 7\cdot12 & 12\cdot3 + 7\cdot7 \end{bmatrix} = \begin{bmatrix} 61 & 36 \\ 144 & 85 \end{bmatrix} \]Step 2: Form Expression
\[ A^2 – 12A – I = \begin{bmatrix} 61 & 36 \\ 144 & 85 \end{bmatrix} – \begin{bmatrix} 60 & 36 \\ 144 & 84 \end{bmatrix} – \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} \]Step 3: Simplify
\[ = \begin{bmatrix} 61 – 60 – 1 & 36 – 36 – 0 \\ 144 – 144 – 0 & 85 – 84 – 1 \end{bmatrix} = \begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix} \]Final Result
\[
A^2 – 12A – I = O
\]
Hence proved.