Question
If \[ A = \begin{bmatrix} 1 & 2 \\ 2 & 1 \end{bmatrix} \] and \( f(x)=x^2-2x-3 \), show that \( f(A)=O \).
Solution
Step 1: Write \(f(A)\)
\[ f(A) = A^2 – 2A – 3I \]Step 2: Compute \(A^2\)
\[ A^2 = \begin{bmatrix} 1 & 2 \\ 2 & 1 \end{bmatrix} \begin{bmatrix} 1 & 2 \\ 2 & 1 \end{bmatrix} = \begin{bmatrix} 1\cdot1 + 2\cdot2 & 1\cdot2 + 2\cdot1 \\ 2\cdot1 + 1\cdot2 & 2\cdot2 + 1\cdot1 \end{bmatrix} = \begin{bmatrix} 5 & 4 \\ 4 & 5 \end{bmatrix} \]Step 3: Form Expression
\[ A^2 – 2A – 3I = \begin{bmatrix} 5 & 4 \\ 4 & 5 \end{bmatrix} – \begin{bmatrix} 2 & 4 \\ 4 & 2 \end{bmatrix} – \begin{bmatrix} 3 & 0 \\ 0 & 3 \end{bmatrix} \]Step 4: Simplify
\[ = \begin{bmatrix} 5 – 2 – 3 & 4 – 4 – 0 \\ 4 – 4 – 0 & 5 – 2 – 3 \end{bmatrix} = \begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix} \]Final Result
\[
f(A) = O
\]
Hence proved.