If `A` and `B` are square matrices of the same order, explain, why in general `(A+B)^2!=A^2+2A B+B^2` (ii) `(A-B)^2!=A^2-2A B+B^2` (iii) `(A+B)(A-B)!=A^2-B^2` .

Explanation

(i)`(A + B)^2 = (A + B)(A + B)`
`(A + B)^2 = A(A + B) + B(A + B)`

` (A + B)^2 = A^2 + AB + BA + B^2`

For the equation, `(A + B)^2 = A^2 + 2AB + B^2` to be valid, we need `AB = BA`.

As the multiplication of two matrices does not satisfy the commutative property in general, `AB != BA`.

Thus, `(A + B)^2 != A^2 + 2AB + B^2`.



(ii)`(A – B)^2 = (A – B)(A – B)`

`(A – B)^2 = A(A – B) – B(A – B)`
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