Question

Let $P=\left[a_{ij}\right]$ be a $3\times 3$ matrix and $Q=\left[b_{ij}\right],$ 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

$2^{13}$

Explanation for the correct option:

Finding the value of determinant $Q$:

Given $P=\left[a_{ij}\right]\text{is a 3×3 matrix}$

And $Q=\left[b_{ij}\right],\text{where}{b}_{ij}=2^{i+j}{a}_{ij}$ for $1\le i,j\le 3$

Let,

$P=\left|\begin{array}{ccc}{a}_{11}& a_{12}& a_{13}\\ a_{21}& a_{22}& a_{23}\\ a_{31}& a_{32}& a_{33}\end{array}\right|=2\dots \dots \left(1\right)$

Now, to write the $Q$ matrix, let's consider

$$\begin{gathered} i=1,j=1\Rightarrow 2^{i+j}=2^2 \\ i=1,j=2\Rightarrow 2^{i+j}=2^3 \\ i=1,j=3\Rightarrow 2^{i+j}=2^4 \end{gathered}$$

Hence, $Q=\left[b_{ij}\right]$ can be written as

$$\begin{gathered} \text{Let}Q=\left[\begin{array}{ccc}{2}^2{a}_{11}& 2^3{a}_{12}& 2^4{a}_{13}\\ 2^3{a}_{21}& 2^4{a}_{22}& 2^5{a}_{23}\\ 2^4{a}_{31}& 2^5{a}_{32}& 2^6{a}_{33}\end{array}\right] \\ \left|Q\right|=\left|\begin{array}{ccc}{2}^2{a}_{11}& 2^3{a}_{12}& 2^4{a}_{13}\\ 2^3{a}_{21}& 2^4{a}_{22}& 2^5{a}_{23}\\ 2^4{a}_{31}& 2^5{a}_{32}& 2^6{a}_{33}\end{array}\right| \\ \left|Q\right|=2^2{2}^3{2}^4\left|\begin{array}{ccc}{a}_{11}& a_{12}& a_{13}\\ 2a_{21}& 2a_{22}& 2a_{23}\\ 2^2{a}_{31}& 2^2{a}_{32}& 2^2{a}_{33}\end{array}\right| \\ \Rightarrow \left|Q\right|=2^9\times 2\times 2^2\left|\begin{array}{ccc}{a}_{11}& a_{12}& a_{13}\\ a_{21}& a_{22}& a_{23}\\ a_{31}& a_{32}& a_{33}\end{array}\right| \\ \Rightarrow \left|Q\right|=2^{12}\left|P\right|\left[\text{from equation}\left(1\right)\right] \\ \Rightarrow \left|Q\right|=2^{13} \end{gathered}$$

Hence option (D) is the correct answer.