Let A = [a_{ij}] and B = [b_{ij}] be two 3 ti | vedclass

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(D) Given $b_{ij} = 3^{(i+j-2)} a_{ji}$.We can write the matrix $B$ as:$B = \begin{bmatrix} a_{11} & 3a_{21} & 9a_{31} \ 3a_{12} & 9a_{22} & 27a_{32}

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$\frac{1}{9}$

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(D) Given $b_{ij} = 3^{(i+j-2)} a_{ji}$.We can write the matrix $B$ as:$B = \begin{bmatrix} a_{11} & 3a_{21} & 9a_{31} \ 3a_{12} & 9a_{22} & 27a_{32} \ 9a_{13} & 27a_{23} & 81a_{33} \end{bmatrix}$Taking common factors from rows and columns:$|B| = (3^0 \cdot 3^1 \cdot 3^2) \cdot \begin{vmatrix} a_{11} & a_{21} & a_{31} \ a_{12} & 3a_{22} & 9a_{32} \ a_{13} & 3a_{23} & 9a_{33} \end{vmatrix}$$|B| = 3^3 \cdot (3^0 \cdot 3^1 \cdot 3^2) \cdot \begin{vmatrix} a_{11} & a_{21} & a_{31} \ a_{12} & a_{22} & a_{32} \ a_{13} & a_{23} & a_{33} \end{vmatrix}$$|B| = 3^3 \cdot 3^3 \cdot |A^T| = 3^6 |A| = 729 |A|$.Given $|B| = 81$,we have $729 |A| = 81$.$|A| = \frac{81}{729} = \frac{1}{9}$.

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

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