Let $A=\left\{a_{i j} ight\}$ be a $3 imes 3$ matrix, where $a_{i j}=\left\{\begin{array}{ccc}(-1

Question

Let $A=\left\{a_{i j}ight\}$ be a $3 	imes 3$ matrix, where $a_{i j}=\left\{\begin{array}{ccc}(-1)^{j-i}, & 	ext { if } & i<j \ 2, & 	ext { if } i=j, \ (-1)^{i+j}, & 	ext { if } i>j\end{array}ight.$ then $\operatorname{det}\left(3 \mathrm{Adj}\left(2 A^{-1}ight)ight)$ is equal to

Professor PC Thomas Classes · Answer

(108) : We have,\ $a_{i j}=\left\{\begin{array}{ccc}(-1)^{j-i}, & 	ext { if } & i<j \ 2, & 	ext { if } & i=j \ (-1)^{i+j}, & 	ext { if } & i>j\end{array}ight.$\ $	herefore \quad A=\left[\begin{array}{ccc}2 & -1 & 1 \ -1 & 2 & -1 \ 1 & -1 & 2\end{array}ight]$\ $\Rightarrow \quad|A|=2(4-1)+1(-2+1)+1(1-2)$\ $=6-1-1=4$\ Now, $\operatorname{det}\left(3 \operatorname{Adj}\left(2 A^{-1}ight)ight)=\left|3 \cdot 22 \operatorname{Adj}\left(A^{-1}ight)ight|$\ $=(12)^{3}\left|\operatorname{Adj}\left(A^{-1}ight)ight|=(12)^{3}\left|A^{-1}ight|^{2}$\ $=\frac{(12)^{3}}{|A|^{2}}=\frac{(12)^{3}}{(4)^{2}}=108$\