Please Solve RD Sharma Class 12 Chapter 6 Adjoint and Inverse of a Matrix  Exercise fill in the blaks Question 17 Maths Textbook Solution.

Answer  $\rightarrow|A| A$

Given  $\rightarrow$ is a non-singular matrix of order $3$

Explanation   $\rightarrow|A|=A . \operatorname{adj} A$                                            …(i)

$(\operatorname{adj} A)^{-1}=\frac{A}{|A|}$

$(\operatorname{adj} A)^{-1}=\frac{\operatorname{adj}(\operatorname{adj} A)}{\operatorname{det}(\operatorname{adj} A)}$                      [$A=a d j(a d j(A)) \text { and } \operatorname{det}(\operatorname{adj} A)=(\operatorname{det} \operatorname{det} A)^{n-1}$ ]

$\operatorname{adj}(\operatorname{adj} A)=(\operatorname{adj} A)^{-1} \operatorname{det}(\operatorname{adj} A)$

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

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

Now, $n= 3$

\begin{aligned} &\operatorname{adj}(\operatorname{adj} A)=(\operatorname{det} A) A \\\\ &=|A| A \end{aligned}