Question

If \(A\) is a square matrix such that \(A^2 = A\), then \((I - A)^3\) is:

• A. \(I - A\)
• B. \(I + A\)
• C. \(I - A^3\)
• D. \(I - A\)

Hint:

For a matrix \(A\) where \(A^2 = A\), it is a projection matrix, and \((I - A)^3 = I - A\).

Answer 1

The Correct Option is D

If \( A \) is a square matrix such that \( A^2 = A \), then \((I - A)^3\) is:

Step 1: Recall the given condition
We are given that \( A^2 = A \). This means that \( A \) is a **idempotent matrix**. An idempotent matrix is a matrix that, when multiplied by itself, results in the same matrix.

Step 2: Consider the expression \( (I - A)^3 \)
We need to compute \( (I - A)^3 \). We can expand this expression step by step. First, let's begin by calculating \( (I - A)^2 \): \[ (I - A)^2 = (I - A)(I - A) = I^2 - 2I \cdot A + A^2 \] Since \( I^2 = I \) and \( A^2 = A \) (from the given condition), we have: \[ (I - A)^2 = I - 2A + A = I - A \]

Step 3: Compute \( (I - A)^3 \)
Now, we can compute \( (I - A)^3 \) using the result from Step 2: \[ (I - A)^3 = (I - A)(I - A)^2 = (I - A)(I - A) \] From Step 2, we know that \( (I - A)^2 = I - A \), so: \[ (I - A)^3 = (I - A)(I - A) = (I - A)^2 = I - A \]

Step 4: Conclusion
Therefore, we conclude that: \[ (I - A)^3 = I - A \]