Question

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

• A. $1 / 9$
• B. $1 / 81$
• C. $1 / 3$
• D. $3$

Answer 1

Answer: (A)

$b_{ij}=\left(3\right)^{\left(i+j-2\right)}a_{ij}$
$B=\begin{bmatrix}a_{11}&3a_{12}&3^{2}a_{13}\\ 3a_{21}&3a_{22}&3a_{23}\\ 3^{2}a_{31}&3^{2}a_{32}&3^{2}a_{33}\end{bmatrix}$
$\Rightarrow \left|B\right|=3\times3^{2}\begin{vmatrix}a_{11}&a_{12}&a_{13}\\ 3a_{21}&3a_{22}&3a_{23}\\ 3^{2}a_{31}&3^{2}a_{32}&3^{2}a_{33}\end{vmatrix}$
$=3^{6}\left|A\right|$
$\Rightarrow \left|A\right|=\frac{81}{27\times27}=\frac{1}{9}$