Question:
If $A$ is a square matrix such that $A^2 = A$, then $(I + A)^3$ is equal to:
If $A$ is a square matrix such that $A^2 = A$, then $(I + A)^3$ is equal to:
Updated On: Mar 29, 2025
- $7A - I$
- $7A + I$
- $7A$
- $I - 7A$
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The Correct Option is B
Solution and Explanation
1. Understand the problem:
Given a square matrix A satisfying \( A^2 = A \), we need to find \( (I + A)^3 \).
2. Expand \( (I + A)^3 \):
Using the binomial expansion:
\[ (I + A)^3 = I^3 + 3I^2A + 3IA^2 + A^3 \]
3. Simplify using \( A^2 = A \):
Since \( A^2 = A \), then \( A^3 = A^2 = A \). Thus:
\[ (I + A)^3 = I + 3A + 3A + A = I + 7A \]
Correct Answer: (B) 7A + I
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