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:
If \( A \) and \( B \) are two non-zero square matrices of the same order such that
\[
(A + B)^2 = A^2 + B^2,
\]
then:
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For square matrices \( A \) and \( B \), the relation \( AB = -BA \) implies the matrices are anti-commutative.
Updated On: Jan 18, 2025
- \( AB = O \)
- \( AB = -BA \)
- \( BA = O \)
- \( AB = BA \)
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The Correct Option is B
Solution and Explanation
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) \( AB = -BA \).
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