Question

If $A$ is a square matrix such that $A^2 = A$, then $(I + A)^3$ is equal to:

• A. $7A - I$
• B. $7A + I$
• C. $7A$
• D. $I - 7A$

Answer 1

The Correct Option is B

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

Answer 2

Given that \( A \) is a square matrix such that \( A^2 = A \). This means \( A \) is an idempotent matrix. We want to find \( (I + A)^3 \).

Let's expand \( (I + A)^3 \):

\[ (I + A)^3 = (I + A)(I + A)(I + A) \] \[ = (I + 2A + A^2)(I + A) \quad \text{(using } (I + A)^2 = I + 2A + A^2\text{)} \] \[ = (I + 2A + A)(I + A) \quad \text{(since } A^2 = A\text{)} \] \[ = (I + 3A)(I + A) \] \[ = I + 3A + A + 3A^2 \] \[ = I + 4A + 3A \quad \text{(since } A^2 = A\text{)} \] \[ = I + 7A \]

Therefore, \( (I + A)^3 = I + 7A \).