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Please Solve RD Sharma Class 12 Chapter 6 Adjoint and Inverse of a Matrix  Exercise fill in the blaks Question 9 Maths Textbook Solution.

Answers (1)

Answer   \rightarrow 5A

Hint    \rightarrow \operatorname{adj}(\operatorname{adj} A)=(\operatorname{det} A)^{n-2} \cdot A

Given  \rightarrow A is an invertible matrix of order 3  and  \left | A \right |=5

Explanation   \rightarrow A^{-1}=\frac{\operatorname{adj} A}{\operatorname{det}(A)}                                                                       …(i)

(\operatorname{adj} A)^{-1}=\left(\operatorname{det}(A) A^{-1}\right)^{-1}=A / \operatorname{det}(A)                             …(ii)

In eqn (i) replace A by adj A

                                      \begin{aligned} &(\operatorname{adj}(A))^{-1}=\frac{\operatorname{adj}(\operatorname{adj}(A))}{\operatorname{det}(\operatorname{adj} A)} \\ &\operatorname{adj}(\operatorname{adj} A)=\operatorname{det}(\operatorname{adj} A)(\operatorname{adj} A)^{-1} \end{aligned}

                                      =(\operatorname{det}(A))^{n-1} \times \frac{A}{\operatorname{det}(A)}                                     [from (ii)]

                                      =(\operatorname{det} A)^{n-2} A

                \operatorname{adj}(\operatorname{adj} A)=|A| A=5 A                                                     [n=3]

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