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:
$(A − B)^2 \neq A^2 − 2AB + B^2$

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Answer

$(A-B)^2-(A-B)(A-B)$
$=A(A-B)-B(A-B)\{\text { using distributive property }\}$
$=A \times A-A B-B A+B \times B$
$=A^2-A B-B A+B^2$
$\neq A^2-2 A B+B^2$
Since, in general matrix multiplication is not commutative $(A B \neq B A)$,
So, $(A-B)^2 \neq A^2-2 A B+B^2$

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