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

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)]

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

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