📘 Question
If a square matrix \(A\) satisfies:
\[
A^2 = I
\]
Find:
\[
(A – I)^3 + (A + I)^3 – 7A
\]
(a) \(A\)
(b) \(I – A\)
(c) \(I + A\)
(d) \(3A\)
✏️ Step-by-Step Solution
Step 1: Use identity
\[ (x-y)^3 + (x+y)^3 = 2x^3 + 6xy^2 \]
Let \(x = A\), \(y = I\)
\[
(A-I)^3 + (A+I)^3 = 2A^3 + 6A
\]
—
Step 2: Use \(A^2 = I\)
\[
A^3 = A \cdot A^2 = A \cdot I = A
\]
\[
2A^3 + 6A = 2A + 6A = 8A
\]
—
Step 3: Substitute
\[
8A – 7A = A
\]
—
✅ Final Answer
\[
\boxed{(a)\; A}
\]
—
💡 Key Concept
If \(A^2 = I\), then higher powers simplify:
\[
A^3 = A
\]
Use algebraic identities to reduce matrix expressions quickly.