Question

If \[ A = \begin{bmatrix} 2 & -1 \\ -1 & 2 \end{bmatrix}, \] such that \[ A^2 - 4A + 3I = 0, \] then \( A^{-1} \) is

• A. \( \frac{-1}{3} \begin{bmatrix} 2 & 1 \\ 1 & 2 \end{bmatrix} \)
• B. \( \frac{-1}{3} \begin{bmatrix} 2 & -1 \\ -1 & 2 \end{bmatrix} \)
• C. \( \frac{1}{3} \begin{bmatrix} -2 & -1 \\ 1 & -2 \end{bmatrix} \)
• D. \( \frac{1}{3} \begin{bmatrix} 2 & 1 \\ 1 & 2 \end{bmatrix} \)

Hint:

When solving for the inverse of a matrix using a given equation, always manipulate the equation to isolate \( A^{-1} \) and use matrix properties to simplify the solution.

Answer 1

The Correct Option is D

Step 1: Use the given equation.
We are given the matrix equation: \[ A^2 - 4A + 3I = 0. \] Rearranging, we get: \[ A^2 = 4A - 3I. \]
Step 2: Find the inverse.
We want to find \( A^{-1} \). Multiply both sides of the equation by \( A^{-1} \): \[ A^{-1} A^2 = A^{-1} (4A - 3I). \] This simplifies to: \[ A = 4I - 3A^{-1}. \] Rearranging to solve for \( A^{-1} \), we get: \[ A^{-1} = \frac{1}{3} \begin{bmatrix} 2 & 1 \\ 1 & 2 \end{bmatrix}. \]
Step 3: Conclusion.
Thus, the correct answer is option (D).