Question

(d) If \( A = \begin{bmatrix} 3 & 1 \\ -1 & 2 \end{bmatrix} \), show that \( A^2 - 5A + 7I = 0 \) and find \( A^{-1} \):

Hint:

Matrix inverses can be found using the adjoint formula or Gaussian elimination.

Answer 1

First, compute \( A^2 \): \[ A^2 = \begin{bmatrix} 3 & 1 \\ -1 & 2 \end{bmatrix} \cdot \begin{bmatrix} 3 & 1 \\ -1 & 2 \end{bmatrix}. \] Then verify: \[ A^2 - 5A + 7I = 0. \] For \( A^{-1} \), use: \[ A^{-1} = \frac{1}{\text{det}(A)} \text{adj}(A). \]