Question

If \( A \) and \( B \) are two non-zero square matrices of the same order such that: $$ (A + B)^2 = A^2 + B^2, $$ then: 

• A. \( AB = O \)
• B. \( AB = -BA \)
• C. \( BA = O \)
• D. \( AB = BA \)
• E.

Hint:

For matrix equations involving expansions like \( (A + B)^2 \), carefully expand and simplify the terms to identify the required relationship between the matrices.

Answer 1

The Correct Option is B

Given the condition: \[ (A + B)^2 = A^2 + B^2. \] Expanding the left-hand side: \[ (A + B)^2 = A^2 + AB + BA + B^2. \] Substituting this into the given equation: \[ A^2 + AB + BA + B^2 = A^2 + B^2. \] Canceling \( A^2 \) and \( B^2 \) from both sides yields: \[ AB + BA = 0. \] Rearranging this equation gives: \[ AB = -BA. \] This demonstrates that \( A \) and \( B \) are anti-commutative, satisfying the relation \( AB = -BA \). Therefore, the correct answer is (B) \( AB = -BA \).