Question

Let $P=\left[a_{ij}\right]$ be a 3 $\times$ 3 matrix and let $Q=\left[b_{ij}\right]$, where $b_{ij}=2^{i+j}{a}_{ij}$ for $1\le i,j\le 3$. If the determinant of P is 2, then the determinant of the matrix Q is

• A. $2^{10}$
• B. $2^{11}$
• C. $2^{12}$
• D. $2^{13}$

Answer 1

The correct option is D $2^{13}$

Given, $\left[P\right]=a_{ij}$ is a order 3 matrix.
Let $Q=\left[\begin{array}{ccc}{2}^2{a}_{11}& 2^3{a}_{12}& 2^4{a}_{13}\\ 2^3{a}_{21}& 2^4{a}_{22}& 2^5{a}_{23}\\ 2^4{a}_{31}& 2^5{a}_{32}& 2^6{a}_{33}\end{array}\right]$
$|Q|=\left|\begin{array}{ccc}{2}^2{a}_{11}& 2^3{a}_{12}& 2^4{a}_{13}\\ 2^3{a}_{21}& 2^4{a}_{22}& 2^5{a}_{23}\\ 2^4{a}_{31}& 2^5{a}_{32}& 2^6{a}_{33}\end{array}\right|$
$\Rightarrow Q=2^2{2}^3{2}^4\left[\begin{array}{ccc}{a}_{11}& a_{12}& a_{13}\\ 2a_{21}& 2a_{22}& 2a_{23}\\ 2^2{a}_{31}& 2^2{a}_{32}& 2^2{a}_{33}\end{array}\right]$
$\Rightarrow |Q|=2^9\times 2\times 2^2\left|\begin{array}{ccc}{a}_{11}& a_{12}& a_{13}\\ a_{21}& a_{22}& a_{23}\\ a_{31}& a_{32}& a_{33}\end{array}\right|$
$\Rightarrow |Q|=2^{12}|P|$
$\Rightarrow |Q|=2^{13}$