Question

If A and B are square matrices such that \( B = -A^{-1}BA \), \(\text{ then }\) \( (A + B)^2 \) is

• A. 0
• B. \( A^2 + B^2 \)
• C. \( A^2 + 2AB + B^2 \)
• D. \( A + B \)

Hint:

To expand matrix expressions like \( (A + B)^2 \), apply standard algebraic expansion rules while remembering the properties of matrices, including inverse and multiplication.

Answer 1

The Correct Option is B

Step 1: The given condition is:
\[ B = -A^{-1}BA \] Step 2: To find \( (A + B)^2 \), expand the expression: \[ (A + B)^2 = A^2 + 2AB + B^2 \] Step 3: Use the given equation \( B = -A^{-1}BA \). Substitute \( B \) into the expansion: \[ (A + (-A^{-1}BA))^2 = A^2 + 2A(-A^{-1}BA) + (-A^{-1}BA)^2 \] Step 4: Simplify the expression: - \( 2A(-A^{-1}BA) = -2AB \) - \( (-A^{-1}BA)^2 = B^2 \) because multiplying a square of a matrix does not change its squared form. Thus, we get: \[ (A + B)^2 = A^2 + B^2 \] Therefore, the correct answer is (b) \( A^2 + B^2 \).