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  4. If A2- A +I=0, then the inverse of A is

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Q. If $A^2- A +I=0$, then the inverse of $A$ is

Matrices

Solution:

If $A$ is any square matrix, then

$AA^1 = I $ and $A^1 I = A^{-1}$

Since, $A^2 - A + 1 = 0 $

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

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

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

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