Question

Let the matrix \[ A=\begin{bmatrix} 3& 1 \\ -1& 2 \end{bmatrix} \] and \[ I=\begin{bmatrix} 1& 0 \\ 0& 1 \end{bmatrix}, \] then \[ A^{2}-5A+7I= \]

• A. \[ \frac{1}{7} \begin{bmatrix} 2&-1 \\ -1& 3 \end{bmatrix} \]
• B. \[ \begin{bmatrix} 1& 1 \\ 1& 1 \end{bmatrix} \]
• C. \[ \begin{bmatrix} 0& 0 \\ 0& 0 \end{bmatrix} \]
• D. \[ \begin{bmatrix} 1& 0 \\ 0& 1 \end{bmatrix} \]

Hint:

For any \(2\times2\) matrix, \[ A= \begin{bmatrix} a& b c& d \end{bmatrix}, \] always compute matrix polynomials in the order: \[ \boxed{A^2\rightarrow kA\rightarrow kI\rightarrow \text{add/subtract}.} \] Also remember the Cayley--Hamilton theorem: \[ A^2-(\operatorname{tr}A)A+(\det A)I=O, \] which often evaluates matrix expressions in one step.

Answer 1

The Correct Option is D