Question

Let $A=\left[a_{ij}\right]$ and $B=\left[b_{ij}\right]$ 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 :

• A. $\frac{1}{9}$
• B. $\frac{1}{81}$
• C. $\frac{1}{3}$
• D. $3$

Answer 1

The correct option is A $\frac{1}{9}$

$b_{ij}=(3{)}^{(i+j-2)}{a}_{ji}$

$B=\left[\begin{array}{ccc}{3}^0{a}_{11}& 3a_{21}& 3^2{a}_{31}\\ 3a_{12}& 3^2{a}_{22}& 3^3{a}_{32}\\ 3^2{a}_{13}& 3^3{a}_{23}& 3^4{a}_{33}\end{array}\right]$

$|B|=\left|\begin{array}{ccc}{3}^0{a}_{11}& 3a_{21}& 3^2{a}_{31}\\ 3a_{12}& 3^2{a}_{22}& 3^3{a}_{32}\\ 3^2{a}_{13}& 3^3{a}_{23}& 3^4{a}_{33}\end{array}\right|$

Taking $3^2$ common each from $C_3$ and $R_3$
$|B|=81\left|\begin{array}{ccc}{a}_{11}& 3a_{21}& a_{31}\\ 3a_{12}& 3^2{a}_{22}& 3a_{32}\\ a_{13}& 3a_{23}& a_{33}\end{array}\right|$

Taking $3$ common each from $C_2$ and $R_2$
$|B|=81\times 9\left|\begin{array}{ccc}{a}_{11}& a_{21}& a_{31}\\ a_{12}& a_{22}& a_{32}\\ a_{13}& a_{23}& a_{33}\end{array}\right|$

Given $|B|=81$
$\Rightarrow 81=81\times 9|A|\Rightarrow |A|=\frac{1}{9}$