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Please solve RD Sharma class 12 chapter Algebra of matrices exercise 4.3 question 69 maths textbook solution

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Answer: Hence proved (A+B)^{2}=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.

Given: A and B be square matrices of same order such that AB=BA

To prove: (A+B)^{2}=A^{2}+2 A B+B^{2}

Now, solving LHS gives

(A+B)^{2}=(A+B)(A+B)     [using distributive of matrix multiplication over addition]

                       \begin{array}{l} =A^{2}+A B+B A+B^{2} \\\\ =A^{2}+A B+A B+B^{2} \quad[\text { as } B A=A B] \\\\ =A^{2}+2 A B+B^{2} \end{array}

Therefore LHS=RHS

Hence, proved.

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