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Explain solution rd sharma class 12 chapter adjoint and inverse of a matrix exercise multiple choice question question 34

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option (a) I

Given – A is a square matrix &A^{2}=A

Hint –Use the concept of square of identity matrix I^{2}=I

\begin{aligned} &\text { Solutions }-(\mathrm{I}-\mathrm{A})^{3}+\mathrm{A} \\ &=(\mathrm{I}-\mathrm{A})^{2}(\mathrm{I}-\mathrm{A})+\mathrm{A} \\ &=\left(\mathrm{I}^{2}-2 \mathrm{AI}+\mathrm{A}^{2}\right)(\mathrm{I}-\mathrm{A})+\mathrm{A} \\ &=(\mathrm{I}-2 \mathrm{~A}+\mathrm{A})(\mathrm{I}-\mathrm{A})+\mathrm{A} \quad \quad\left(\therefore \mathrm{A}^{2}=\mathrm{A} \text { and } \mathrm{I}^{2}=\mathrm{I}\right) \end{aligned}

\begin{aligned} &=(I-A)(I-A)+A \\ &=\left(I^{2}-2 A I+A^{2}\right)+A \\ &=(I-2 A+A)+A \quad \quad\left(\therefore A^{2}=A \text { and } I^{2}=I\right) \\ &=I-A+A=I \\ &\Rightarrow(I-A)^{3}+A=I \end{aligned} 

Hence, option (a) is correct

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