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Explain solution for rd sharma class class 12 chapter Algebra of matrices exercise 4.3 question 30 math

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Answer: (2+A)^{3}-19 A=A^{2}-A+8

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 is a square matrix A^2 = A

Consider: (2+A)^{3}-19 A

As we know (a+b)^{3}=a^{3}+b^{3}+3 a^{2} b+3 a b^{2}

Then,\Rightarrow(2+A)^{3}-19 A\\ =8+A^{3}+3(2)^{2} A+3(2) A^{2}-19 A\\ =8+A^{3}+12 A+6 A^{2}-19 A\\ =A^{3}+6 A^{2}-7 A+8\\ =A^{2} A+6(A)-7(A)+8 \quad\left[\right. As \ \ given \left.A^{2}=A\right]\\ =A A+6 A-7 A+8 \quad\left[\right.As \ \ given \left.A^{2}=A\right]\\ =A^{2}-A+8

Hence, (2+A)^{3}-19 A=A^{2}-A+8

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