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. \( AB = BA \)

Hint:

For square matrices \( A \) and \( B \), the relation \( AB = -BA \) implies the matrices are anti-commutative.

Answer 1

The Correct Option is B

Expand the left-hand side of the given equation: \[ (A + B)^2 = A^2 + AB + BA + B^2. \] 

Equating both sides: \[ A^2 + AB + BA + B^2 = A^2 + B^2. \] 

Cancel \( A^2 \) and \( B^2 \): \[ AB + BA = 0. \]

Rearranging: \[ AB = -BA. \] 

Therefore, the correct answer is (B) \( AB = -BA \).