Question
If \[ f(x)=x^3+4x^2-x \] and \[ A = \begin{bmatrix} 0 & 1 & 2 \\ 2 & -3 & 0 \\ 1 & -1 & 0 \end{bmatrix} \] find \(f(A)\).
Solution
Step 1: Write Expression
\[ f(A)=A^3 + 4A^2 – A \]Step 2: Compute \(A^2\)
\[ A^2 = \begin{bmatrix} 4 & -5 & 0 \\ -6 & 11 & 4 \\ -2 & 4 & 2 \end{bmatrix} \]Step 3: Compute \(A^3 = A^2 \cdot A\)
\[ A^3 = \begin{bmatrix} -10 & 19 & 8 \\ 26 & -43 & -12 \\ 10 & -16 & -4 \end{bmatrix} \]Step 4: Form Expression
\[ A^3 + 4A^2 – A = \begin{bmatrix} -10 & 19 & 8 \\ 26 & -43 & -12 \\ 10 & -16 & -4 \end{bmatrix} + \begin{bmatrix} 16 & -20 & 0 \\ -24 & 44 & 16 \\ -8 & 16 & 8 \end{bmatrix} – \begin{bmatrix} 0 & 1 & 2 \\ 2 & -3 & 0 \\ 1 & -1 & 0 \end{bmatrix} \]Step 5: Simplify
\[ = \begin{bmatrix} 6 & -2 & 6 \\ 0 & 4 & 4 \\ 1 & 1 & 4 \end{bmatrix} \]Final Answer
\[
f(A)=
\begin{bmatrix}
6 & -2 & 6 \\
0 & 4 & 4 \\
1 & 1 & 4
\end{bmatrix}
\]