Question
If $A$ and $B$ are square matrices of the same order, explain, why in general:
$(A + B)^2 \neq A^2 + 2AB + B^2$
$(A + B)^2 \neq A^2 + 2AB + B^2$
✓
Answer
LHS $= (A + B)^2$
$= (A + B)(A + B)$
$= A(A + B) + B(A + B)$
$= A^2 + AB + BA +B^2$
We know that a matrix does not have commutative property. So,
$AB \neq BA$
Thus,
$(A + B)^2 \neq A^2 + 2AB + B^2$
$= (A + B)(A + B)$
$= A(A + B) + B(A + B)$
$= A^2 + AB + BA +B^2$
We know that a matrix does not have commutative property. So,
$AB \neq BA$
Thus,
$(A + B)^2 \neq A^2 + 2AB + B^2$
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