Let A and B be square matrices of the same order. Does

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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