Question

A square matrix (non-singular) satisfies $A^2 - A + 2I = 0$. Then $A^{-1}=$

• A. $\dfrac{I-A}{2}$
• B. $I-A$
• C. $\dfrac{I+A}{2}$
• D. $I+A$

Hint:

When a matrix satisfies a polynomial equation, multiply by $A^{-1}$ to express the inverse in terms of $A$ and $I$.

Answer 1

The Correct Option is A

Step 1: Start with the given matrix equation: \[ A^2 - A + 2I = 0 \]
Step 2: Rearrange the equation to isolate $A^2$: \[ A^2 = A - 2I \]
Step 3: Multiply both sides by $A^{-1}$ (since $A$ is non-singular): \[ A = I - 2A^{-1} \]
Step 4: Rearrange to solve for $A^{-1}$: \[ 2A^{-1} = I - A \]
Step 5: Divide both sides by 2: \[ A^{-1} = \frac{I-A}{2} \]