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Singular and Non Singualar Matrices

A square non-...

Question

A square non-singular matrix A satisfies $A^{2} - A + 2 I = 0$ , then $A^{- 1} =$

I - A

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\frac{1}{2} (I - A)

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I + A

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\frac{1}{2} (I + A)

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Solution

The correct option is D $\frac{1}{2} (I - A)$
Given, $A^{2} - A + 2 I = 0$

$\Rightarrow A^{2} A^{- 1} - A A^{- 1} + 2 I A^{- 1} = 0$

$\Rightarrow A - I + 2 A^{- 1} = 0$

$\Rightarrow 2 A^{- 1} = I - A$

$\Rightarrow A^{- 1} = \frac{1}{2} (I - A)$

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