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If A is a non singular matrix and

$\left (I - A + A^{2} - ..... + (-1)^{n}A^{n}\right ) =0$)

Then $A^{-1}$  is
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$\left ( I-A+A^{2}-\cdots+(-1)^{n}A^{n} \right )=0$

$I=\left ( A-A^{2}+A^{3}\cdots +(-1)^{n+1}A^{n+1} \right )$

Multiply both size by $A^{-1}$

$A^{-1}=\left ( I-A+A^{2}\cdots +(-1)^{n+1}A^{n} \right )$
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it should be A−1=(I−A+A2⋯+(−1)n+1 An-1)
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