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Question

Question: If \(A^{2} + A - I = 0,thenA^{- 1} =\)...

If A2+A−I=0,thenA−1=A^{2} + A - I = 0,thenA^{- 1} =

A

A−IA - I

B

I−AI - A

C

I+AI + A

D

None of these

Answer

I+AI + A

Explanation

Solution

A2+A−I=0⇒A−1(A2+A−I)=A−1.0A^{2} + A - I = 0 \Rightarrow A^{- 1}\left( A^{2} + A - I \right) = A^{- 1}.0 ⇒A+A−1A−A−1I=0\Rightarrow A + A^{- 1}A - A^{- 1}I = 0 ⇒(A+I)−A−1=0⇒A−1=A+I\Rightarrow (A + I) - A^{- 1} = 0 \Rightarrow A^{- 1} = A + I