Question

If A is a square matrix and \( A^2 = A \), then \( (A + I)^3 - 7A \) will be:

• A. A
• B. 3A
• C. I - A
• D. I

Hint:

When simplifying matrix polynomials, always use the given conditions (\( A^2=A \), \( A^2=I \), etc.) to reduce higher powers of the matrix A to simpler forms (like A or I).

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
We need to simplify a matrix expression given a specific property of the matrix A (\( A^2 = A \), which means A is an idempotent matrix). We use the properties of matrix algebra, including the binomial expansion for matrices that commute (A and I commute).
Step 2: Key Formula or Approach:
- Binomial expansion: \( (X+Y)^3 = X^3 + 3X^2Y + 3XY^2 + Y^3 \).
- Identity matrix properties: \( I^n = I \) for any positive integer n, and \( AI = IA = A \).
- Given property: \( A^2 = A \). This also implies \( A^3 = A^2 \cdot A = A \cdot A = A^2 = A \).
Step 3: Detailed Explanation or Calculation:
First, expand \( (A + I)^3 \):
\[ (A + I)^3 = A^3 + 3A^2I + 3AI^2 + I^3 \] Using the identity matrix properties, this simplifies to:
\[ (A + I)^3 = A^3 + 3A^2 + 3A + I \] Now, use the given condition \( A^2 = A \). This implies \( A^3 = A \).
Substitute these into the expansion:
\[ (A + I)^3 = (A) + 3(A) + 3A + I \] \[ (A + I)^3 = 7A + I \] Now, substitute this result back into the original expression:
\[ (A + I)^3 - 7A = (7A + I) - 7A \] \[ = 7A - 7A + I = I \] Step 4: Final Answer:
The expression simplifies to the identity matrix, I.