📘 Question
If a square matrix \(A\) satisfies:
\[
A^2 = A
\]
Find the value of:
\[
(I + A)^3 – 7A
\]
(a) \(A\)
(b) \(I – A\)
(c) \(I\)
(d) \(3A\)
✏️ Step-by-Step Solution
Step 1: Expand \((I + A)^3\)
\[
(I + A)^3 = I + 3A + 3A^2 + A^3
\]
Step 2: Use \(A^2 = A\)
\[
A^2 = A,\quad A^3 = A
\]
\[
(I + A)^3 = I + 3A + 3A + A
\]
\[
= I + 7A
\]
Step 3: Substitute
\[
(I + A)^3 – 7A = (I + 7A) – 7A
\]
\[
= I
\]
✅ Final Answer
\[
\boxed{(c)\; I}
\]
💡 Key Concept
If \(A^2 = A\), then \(A\) is an idempotent matrix. This simplifies higher powers like \(A^3 = A\), making expressions easy to evaluate.