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  4. If a square matrix satisfies the relation A2 + A - I = 0 then A-1:

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Q. If a square matrix satisfies the relation $A^2 + A - I = 0$ then $A^{-1}$:

Determinants

Solution:

Given, $A^2 + A - I = 0$
Pre Multiply it with $A^{-1}$, we get
$A + I - A^{-1} = 0 $
$\Rightarrow \, A^{-1} = I + A$
Hence, $A^{-1}$ exists and equals I + A.