Let $A = [a_{ij}]$ and $B = [b_{ij}]$ be two $3 imes 3$ real matrices such that $b_{ij} = 3^{i+j-2}a_{ij}$, where $i, j = 1, 2, 3$. If the determinant of $B$ is $81$, then the determinant of $A$ is

Question

Let $A = [a_{ij}]$ and $B = [b_{ij}]$ be two $3 	imes 3$ real matrices such that $b_{ij} = 3^{i+j-2}a_{ij}$, where $i, j = 1, 2, 3$. If the determinant of $B$ is $81$, then the determinant of $A$ is

Aicharya · Accepted Answer

$|B| = \begin{aligned} &\begin{vmatrix} b_{11} & b_{12} & b_{13} \ b_{21} & b_{22} & b_{23} \ b_{31} & b_{32} & b_{33} \end{vmatrix} \ &= \begin{vmatrix} 3^{0}a_{11} & 3^{1}a_{12} & 3^{2}a_{13} \ 3^{1}a_{21} & 3^{2}a_{22} & 3^{3}a_{23} \ 3^{2}a_{31} & 3^{3}a_{32} & 3^{4}a_{33} \end{vmatrix} \end{aligned}$ $\Rightarrow 81 = 3^{3} \cdot 3 \cdot 3^{2} |A| \Rightarrow 3^{4} = 3^{6} |A| \Rightarrow |A| = \frac{1}{9}$

Let $A = [a_{ij}]$ and $B = [b_{ij}]$ be two $3 \times 3$ real matrices such that $b_{ij} = 3^{i+j-2}a_{ij}$…

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