MCQ
A square, non-singular matrix A satisfies $A ^2- A +2 I =0$, then $A ^{-1}$ is equal to
- A$I - A$
- ✓$\frac{1}{2}( I - A )$
- C$\frac{1}{2}( I + A )$
- D$I + A$
✓
Answer
Correct option: B.
$\frac{1}{2}( I - A )$
(B) $A^2-A+2 I=0$
$\Rightarrow A \cdot A - A +2 I =0$
$\Rightarrow A ^{-1} \cdot A \cdot A - A ^{-1} \cdot A+2 A^{-1} \cdot I =0$
$\Rightarrow A - I +2 A^{-1}=0$
$\Rightarrow 2 A^{-1}= I - A$
$\Rightarrow A ^{-1}=\frac{1}{2}( I - A )$
$\Rightarrow A \cdot A - A +2 I =0$
$\Rightarrow A ^{-1} \cdot A \cdot A - A ^{-1} \cdot A+2 A^{-1} \cdot I =0$
$\Rightarrow A - I +2 A^{-1}=0$
$\Rightarrow 2 A^{-1}= I - A$
$\Rightarrow A ^{-1}=\frac{1}{2}( I - A )$
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