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Provide solution for rd sharma math class 12 chapter Algebra of matrices exercise 4.3 question 67 sub question (ii)

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Answer: In general matrix multiplication is not always commutative(A B \neq B A)

so, (A-B)^{2} \neq A^{2}-2 A B+B^{2}

Hint: Matrix multiplication is only possible, when the number of columns of first matrix is equal to the number of rows of second matrix.

(x-y)^{2}=x^{2}+y^{2}-2 x y

Given: A and B be square matrices of same order (A-B)^{2} \neq A^{2}-2 A B+B^{2}

(A-B)^{2}=(A-B)(A-B) \\\\

                        =A(A-B)-B(A-B) \\[using distributive property]

                       \\=A \times A-A B-B A+B \times B \\\\ =A^{2}-A B-B A+B^{2} \\\\ \neq A^{2}-2 A B+B^{2}

Since, in general matrix multiplication is not always commutative A B \neq BA,

So, (A-B)^{2} \neq A^{2}-2 A B+B^{2}

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