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

We are given the condition: \[ (A + B)^2 = A^2 + B^2. \] Expand the left-hand side using the distributive property of matrix multiplication: \[ (A + B)^2 = A^2 + AB + BA + B^2. \] Substitute this into the equation: \[ A^2 + AB + BA + B^2 = A^2 + B^2. \] Cancel \( A^2 \) and \( B^2 \) from both sides: \[ AB + BA = 0. \] Rearrange: \[ AB = -BA. \] This shows that \( A \) and \( B \) satisfy the relation \( AB = -BA \), meaning \( A \) and \( B \) are anti-commutative. Hence, the correct answer is (B) \( AB = -BA \).