📘 Question
If a square matrix \( A \) satisfies \( A^2 = A \), find the value of:
\[
7A – (I + A)^3
\]
where \( I \) is the identity matrix.
✏️ Step-by-Step Solution
Given:
\[
A^2 = A
\]
This means \(A\) is an idempotent matrix.
Step 1: Expand \((I + A)^3\)
\[
(I + A)^3 = I^3 + 3I^2A + 3IA^2 + A^3
\]
Step 2: Use properties
- \(I^2 = I\)
- \(IA = A\)
- \(A^2 = A\)
- \(A^3 = A\)
\[
(I + A)^3 = I + 3A + 3A + A
\]
\[
= I + 7A
\]
Step 3: Substitute
\[
7A – (I + A)^3 = 7A – (I + 7A)
\]
Step 4: Simplify
\[
= 7A – I – 7A
\]
\[
= -I
\]
✅ Final Answer
\[
\boxed{-I}
\]
💡 Key Concept
A matrix satisfying \(A^2 = A\) is called an idempotent matrix. This property simplifies higher powers like \(A^3 = A\), making problems much easier to solve.