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  4. If A=[1 2 3 1 3 5 2 1 6], then ( textAdj( textAdj A))-1=

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Q. If $A=\begin{bmatrix}1 & 2 & 3 \\ 1 & 3 & 5 \\ 2 & 1 & 6\end{bmatrix}$, then $(\text{Adj}(\text{Adj}\, A))^{-1}=$

TS EAMCET 2019

Solution:

$A=\begin{bmatrix}1 & 2 & 3 \\ 1 & 3 & 5 \\ 2 & 1 & 6\end{bmatrix}$
$|A|=1(18-5)-2(6-10)+3(1-6)$
$=13+8-15=6$
$\text{Adj} A=\begin{bmatrix}13 & -9 & 1 \\ 4 & 0 & -2 \\ -5 & 3 & 1\end{bmatrix}$
$A^{-1}=\frac{\text{Adj} A}{|A|}=\frac{1}{6}\begin{bmatrix}13 & -9 & 1 \\ 4 & 0 & -2 \\ -5 & 3 & 1\end{bmatrix}$
Now, $\text{Adj}(\text{Adj} A)=|A| A$
$\text{Adj}(\text{Adj} A)=6 A$
$(\text{Adj}(\text{Adj} A))^{-1}=(6 A)^{-1}$
$=\frac{1}{36}\begin{bmatrix}13 & -9 & 1 \\ 4 & 0 & -2 \\ -5 & 3 & 1\end{bmatrix}$
$\therefore (\text{Adj}(\text{Adj} A))^{-1}=\frac{1}{36}\begin{bmatrix}13 & -9 & 1 \\ 4 & 0 & -2 \\ -5 & 3 & 1\end{bmatrix}$