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Question
Let A and B be square matrices of the same order. Does (A + B)2 = A2 + 2AB + B2 hold? If not, why?
Solution
\[LHS = \left( A + B \right)^2 \]
\[ = \left( A + B \right)\left( A + B \right)\]
\[ = A\left( A + B \right) + B\left( A + B \right)\]
\[ = A^2 + AB + BA + B^2\]
We know that a matrix does not have commutative property. So,
AB ≠ BA
Thus,
\[\left( A + B \right)^2\]≠
\[A^2 + 2AB + B^2\]
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