Question
If \[ f(x)=x^2-2x \] and \[ A = \begin{bmatrix} 0 & 1 & 2 \\ 4 & 5 & 0 \\ 0 & 2 & 3 \end{bmatrix} \] find \(f(A)\).
Solution
Step 1: Write Expression
\[ f(A)=A^2 – 2A \]Step 2: Compute \(A^2\)
\[ A^2 = \begin{bmatrix} 4 & 9 & 6 \\ 20 & 29 & 8 \\ 8 & 16 & 9 \end{bmatrix} \]Step 3: Form Expression
\[ A^2 – 2A = \begin{bmatrix} 4 & 9 & 6 \\ 20 & 29 & 8 \\ 8 & 16 & 9 \end{bmatrix} – \begin{bmatrix} 0 & 2 & 4 \\ 8 & 10 & 0 \\ 0 & 4 & 6 \end{bmatrix} \]Step 4: Simplify
\[ = \begin{bmatrix} 4 & 7 & 2 \\ 12 & 19 & 8 \\ 8 & 12 & 3 \end{bmatrix} \]Final Answer
\[
f(A)=
\begin{bmatrix}
4 & 7 & 2 \\
12 & 19 & 8 \\
8 & 12 & 3
\end{bmatrix}
\]