If $A^2 = A$, then $(A + I)^5$ is equal to (where $I$ is identity matrix)
Step-by-Step Solution
Key Concept: When A² = A (idempotent matrix), all higher powers reduce to A itself (A³ = A⁴ = A⁵ = ... = A). Apply binomial expansion to (A + I)⁵ and use this reduction property to simplify.
<div>$(A + I)^5 = I + 5A + 10A^2 + 10A^3 + 5A^4 + A^5$<br/>$= I + 5A + 10A + 10A + 5A + A$<br/>$(\because A^2 = A \Rightarrow A^3 = A^4 = A^5 = \dots = A)$<br/>$= I + 31A$</div>
Correct Answer: C