If A and B are square matrices of the same order, explain, why in… - Vidyadip

Question

If A and B are square matrices of the same order, explain, why in general:
(A − B)² ≠ A² − 2AB + B²

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Answer

(A - B)² - (A - B)(A - B)
= A(A - B) - B(A - B) {using distributive property}
= A × A - AB - BA + B × B
= A² - AB - BA + B²
≠ A² - 2AB + B²
Since, in general matrix multiplication is not commutative (AB ≠ BA),
So, (A - B)² ≠ A² - 2AB + B²

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