Let $A=\left[a_{i j}\right]$ and $B=\left[b_{i j}\right]$ be two $3 \times 3$ real matrices such that $b_{i j}=(3)^{(i+j-2)} a_{i j}$, where $i, j=1,2,3$. If the determinant of $B$ is 81 , then the determinant of $A$ is:
Correct Option: , 4
It is given that $|B|=81$
$\therefore \quad|B|=\left|\begin{array}{lll}b_{11} & b_{12} & b_{13} \\ b_{21} & b_{22} & b_{23} \\ b_{31} & b_{32} & b_{33}\end{array}\right|=\left|\begin{array}{lll}3^{0} a_{11} & 3^{1} a_{12} & 3^{2} a_{13} \\ 3^{1} a_{21} & 3^{2} a_{22} & 3^{3} a_{23} \\ 3^{2} a_{31} & 3^{3} a_{32} & 3^{4} a_{33}\end{array}\right|$
$\Rightarrow 81=3^{3} \cdot 3^{2} \cdot 3^{1}|A|$
$\Rightarrow 3^{4}=3^{6}|A| \Rightarrow|A|=\frac{1}{9}$
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