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

Answer 1

None of these

\[I - A^5 = \left( I - A \right)\left( I + A + A^2 + A^3 + A^4 \right)\]

Now,

\[ A^5 = 0\]

\[ \Rightarrow I = \left( I - A \right)\left( I + A + A^2 + A^3 + A^4 \right)\]

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

\[ \Rightarrow \left( I - A \right)^{- 1} = I + A + A^2 + A^3 + A^4\]