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Question
If A5 = O such that \[A^n \neq I\text{ for }1 \leq n \leq 4,\text{ then }\left( I - A \right)^{- 1}\] equals ________ .
Options
A4
A3
I + A
none of these
Solution
None of these
\[I - A^5 = \left( I - A \right)\left( I + A + A^2 + A^3 + A^4 \right)\]
Now,
\[ A^5 = 0\]
\[ \Rightarrow I = \left( I - A \right)\left( I + A + A^2 + A^3 + A^4 \right)\]
\[ \Rightarrow \frac{I}{\left( I - A \right)} = \left( I + A + A^2 + A^3 + A^4 \right)\]
\[ \Rightarrow \left( I - A \right)^{- 1} = I + A + A^2 + A^3 + A^4\]
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