Question
If \[ A = \begin{bmatrix} 1 & 2 & 0 \\ 3 & -4 & 5 \\ 0 & -1 & 3 \end{bmatrix} \] compute \[ A^2 – 4A + 3I_3. \]
Solution
Step 1: Compute \(A^2\)
\[ A^2 = \begin{bmatrix} 7 & -6 & 10 \\ -9 & 17 & -5 \\ -3 & 1 & 4 \end{bmatrix} \]Step 2: Form Expression
\[ A^2 – 4A + 3I_3 = \begin{bmatrix} 7 & -6 & 10 \\ -9 & 17 & -5 \\ -3 & 1 & 4 \end{bmatrix} – \begin{bmatrix} 4 & 8 & 0 \\ 12 & -16 & 20 \\ 0 & -4 & 12 \end{bmatrix} + \begin{bmatrix} 3 & 0 & 0 \\ 0 & 3 & 0 \\ 0 & 0 & 3 \end{bmatrix} \]Step 3: Simplify
\[ = \begin{bmatrix} 7 – 4 + 3 & -6 – 8 + 0 & 10 – 0 + 0 \\ -9 – 12 + 0 & 17 + 16 + 3 & -5 – 20 + 0 \\ -3 – 0 + 0 & 1 + 4 + 0 & 4 – 12 + 3 \end{bmatrix} \] \[ = \begin{bmatrix} 6 & -14 & 10 \\ -21 & 36 & -25 \\ -3 & 5 & -5 \end{bmatrix} \]Final Answer
\[
A^2 – 4A + 3I_3 =
\begin{bmatrix}
6 & -14 & 10 \\
-21 & 36 & -25 \\
-3 & 5 & -5
\end{bmatrix}
\]