Question

If \( A \) is a square matrix satisfying the equation \( A^2 - 5A + 7I = 0 \), where \( I \) is the identity matrix and \( 0 \) is the null matrix of the same order, then \( A^{-1} \) is:

• A. \( \frac{1}{7} (A - 5I) \)
• B. \( 7(5I - A) \)
• C. \( \frac{1}{5}(7I - A) \)
• D. \( \frac{1}{7}(5I - A) \)

Hint:

When solving matrix equations, try to isolate the matrix you're looking for by multiplying both sides of the equation by the inverse or other relevant operations.

Answer 1

The Correct Option is D

Given the equation \( A^2 - 5A + 7I = 0 \), multiply both sides by \( A^{-1} \): \[ A^{-1}(A^2 - 5A + 7I) = A^{-1}(0) \] \[ A^{-1}A^2 - 5A^{-1}A + 7A^{-1}I = 0 \] \[ A - 5I + 7A^{-1} = 0 \] \[ 7A^{-1} = 5I - A \] \[ A^{-1} = \frac{1}{7}(5I - A) \] Thus, the correct answer is \( A^{-1} = \frac{1}{7}(5I - A) \).