Question:
Let $A$ and $B$ be square matrices of the same order. Does $(A+B)^{2}=A^{2}+2 A B+B^{2}$ hold? If not, why?
Solution:
LHS $=(A+B)^{2}$
$=(A+B)(A+B)$
$=A(A+B)+B(A+B)$
$=A^{2}+A B+B A+B^{2}$
We know that a matrix does not have commutative property. So,
AB ≠ BA
Thus,
$(A+B)^{2} \neq A^{2}+2 A B+B^{2}$
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