Question
If \[ A = \begin{bmatrix} 1 & 0 & 2 \\ 0 & 2 & 1 \\ 2 & 0 & 3 \end{bmatrix} \] show that \(A\) is a root of the polynomial \[ f(x)=x^3-6x^2+7x+2. \]
Solution
Step 1: Write \(f(A)\)
\[ f(A)=A^3 – 6A^2 + 7A + 2I \]Step 2: Compute \(A^2\)
\[ A^2 = \begin{bmatrix} 5 & 0 & 8 \\ 2 & 4 & 5 \\ 8 & 0 & 13 \end{bmatrix} \]Step 3: Compute \(A^3 = A^2 \cdot A\)
\[ A^3 = \begin{bmatrix} 21 & 0 & 34 \\ 12 & 8 & 21 \\ 34 & 0 & 55 \end{bmatrix} \]Step 4: Form Expression
\[ A^3 – 6A^2 + 7A + 2I = \begin{bmatrix} 21 & 0 & 34 \\ 12 & 8 & 21 \\ 34 & 0 & 55 \end{bmatrix} – \begin{bmatrix} 30 & 0 & 48 \\ 12 & 24 & 30 \\ 48 & 0 & 78 \end{bmatrix} + \begin{bmatrix} 7 & 0 & 14 \\ 0 & 14 & 7 \\ 14 & 0 & 21 \end{bmatrix} + \begin{bmatrix} 2 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 2 \end{bmatrix} \]Step 5: Simplify
\[ = \begin{bmatrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{bmatrix} \]Final Result
\[
f(A)=O
\]
Hence, \(A\) is a root of the given polynomial.