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Multiplication of Matrices

lf A is non...

Question

lf $A$ is non-singular and $(A + I) (A - 3 I) = 0$ then $3 A^{- 1} - A + 2 I$

I

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0

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2 I

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6 I

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Solution

The correct option is B $0$
We have,
$(A + I) (A - 3 I) = 0$
$\Rightarrow A^{2} - 3 A + A - 3 I = 0 [∵ A I = I A = A, I^{2} = I]$
$\Rightarrow A^{2} - 2 A - 3 I = 0$
Multiply both sides by $A^{- 1}$
$\Rightarrow A - 2 I - 3 A^{- 1} = 0 [∵ A A^{- 1} = I]$
$\Rightarrow 3 A^{- 1} - A + 2 I = 0$

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