Question

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

Hint:

When dealing with matrix equations, leverage properties like idempotence (i.e., \( A^2 = A \)) to simplify and expand expressions effectively.

Answer 1

Step 1: Given that \( A^2 = A \), this means that \( A \) is a idempotent matrix. 

Step 2: Let’s first expand \( (I - A)^3 \) using the binomial expansion: \[ (I - A)^3 = I^3 - 3I^2A + 3IA^2 - A^3 = I - 3A + 3A - A = I - A. \] Thus: \[ (I - A)^3 = I - A. \] 

Step 3: Now, expand \( (I + A)^2 \): \[ (I + A)^2 = I^2 + 2IA + A^2 = I + 2A + A = I + 3A. \] 

Step 4: Subtract \( (I + A)^2 \) from \( (I - A)^3 \): \[ (I - A)^3 - (I + A)^2 = (I - A) - (I + 3A) = I - A - I - 3A = -4A. \] 

Step 5: Now, factorize the result: \[ -4A = 2(I - 2A). \] 

Thus, the final expression is: \[ (I - A)^3 - (I + A)^2 = 2(I - 2A). \]