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

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

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