Question

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

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

Answer 1

The correct option is A

$\frac{1}{9}$

Explanation for the correct option:

Step:1 Solving the determinant value of matrix $B$.

$\begin{array}{rcl}{b}_{ij}& =& \left(3^{i+j-2}\right)a_{ji}\\ B& =& \left[\begin{array}{ccc}{3}^0{a}_{11}& 3^1{a}_{21}& 3^2{a}_{31}\\ 3^1{a}_{12}& 3^2{a}_{22}& 3^3{a}_{32}\\ 3^2{a}_{13}& 3^3{a}_{23}& 3^4{a}_{33}\end{array}\right]\\ \left|B\right|& =& \left|\begin{array}{ccc}{a}_{11}& 3a_{21}& 3^2{a}_{31}\\ 3a_{12}& 3^2{a}_{22}& 3^3{a}_{32}\\ 3^2{a}_{13}& 3^3{a}_{23}& 3^4{a}_{33}\end{array}\right|\\ & =& 3^2\times \left|\begin{array}{ccc}{a}_{11}& 3^1{a}_{21}& a_{31}\\ 3^1{a}_{12}& 3^2{a}_{22}& 3a_{32}\\ 3^2{a}_{13}& 3^3{a}_{23}& 3^2{a}_{33}\end{array}\right|\\ & =& 3^2\times 3^2\times \left|\begin{array}{ccc}{a}_{11}& 3^1{a}_{21}& a_{31}\\ 3^1{a}_{12}& 3^2{a}_{22}& 3a_{32}\\ a_{13}& 3a_{23}& a_{33}\end{array}\right|\\ & =& 3^2\times 3^2\times 3\left|\begin{array}{ccc}{a}_{11}& a_{21}& a_{31}\\ 3^1{a}_{12}& 3a_{22}& 3a_{32}\\ a_{13}& a_{23}& a_{33}\end{array}\right|\\ & =& 3^2\times 3^2\times 3\times 3\times \left|\begin{array}{ccc}{a}_{11}& a_{21}& a_{31}\\ 3^1{a}_{12}& 3a_{22}& 3a_{32}\\ a_{13}& a_{23}& a_{33}\end{array}\right|\\ & =& 81\times 9\times \left|\begin{array}{ccc}{a}_{11}& a_{21}& a_{31}\\ 3^1{a}_{12}& 3a_{22}& 3a_{32}\\ a_{13}& a_{23}& a_{33}\end{array}\right|\end{array}$

Step:2 Compare the determinant values of matrix $B$with matrix $A$:

$$\begin{gathered} \Rightarrow \left|B\right|=81\times 9\times \left|A\right| \\ \Rightarrow 81=81\times 9\times \left|A\right| \\ \Rightarrow \left|A\right|=\frac{1}{9} \end{gathered}$$

Therefore Option(A) is correct answer.