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  • Provide Solutio for RD Sharma Class 12 Chapter 6 Adjoint and Inverse Matrix Exercise Fill in the blanks Question 20

Provide Solutio for RD Sharma Class 12 Chapter 6 Adjoint and Inverse Matrix Exercise Fill in the blanks Question 20

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Answer   \rightarrow I

Given   \rightarrow A A^{T}=A^{T} A \text { and } B=A^{-1} A^{T}

Explanation    \rightarrow B=A^{-1} A^{T}

                   B^{-1}=\left(A^{-1} A^{T}\right)^{-1}

                    =\left(A^{T}\right)^{-1}\left(A^{-1}\right)^{-1} \qquad\left[(A B)^{-1}=B^{-1} A^{-1}\right]

Now,

               \begin{aligned} &B B^{-1}=A^{-1} A^{T}\left(A^{T}\right)^{-1}\left(A^{-1}\right)^{-1} \\ &=A^{-1} A^{T}\left(A^{T}\right)^{-1} A \\ &=A^{-1}\left(A A^{-1}\right)^{T} A \\ &=A^{-1} \cdot I \cdot A \end{aligned}

                =A^{-1} A=1             

                                      B B^{-1}=I 

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