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Question
If A and B are square matrices of the same order, explain, why in general
(A + B) (A − B) ≠ A2 − B2
Solution
\[\left( iii \right) LHS = \left( A + B \right)\left( A - B \right)\]
\[ = A\left( A - B \right) + B\left( A - B \right)\]
\[ = A^2 - AB + BA - B^2\]
We know that a matrix does not have commutative property. So,
AB ≠ BA
Thus,
\[\left( A + B \right)\left( A - B \right)\]≠
\[A^2 - B^2\]
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