Solve this

Question:

If $A^{5}=O$ such that $A^{n} \neq I$ for $1 \leq n \leq 4$, then $(I-A)^{-1}$ equals

(a) $A^{4}$

(b) $A^{3}$

(c) $l+A$

(d) none of these

Solution:

(d) none of the these

$I-A^{5}=(I-A)\left(I+A+A^{2}+A^{3}+A^{4}\right)$

Now,

$A^{5}=0$

$\Rightarrow I=(I-A)\left(I+A+A^{2}+A^{3}+A^{4}\right)$

$\Rightarrow \frac{I}{(I-A)}=\left(I+A+A^{2}+A^{3}+A^{4}\right)$

$\Rightarrow(I-A)^{-1}=I+A+A^{2}+A^{3}+A^{4}$

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