Matrices and Determinants 2 Question 13
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15. Let $P=\left[a _{i j}\right]$ be a $3 \times 3$ matrix and let $Q=\left[b _{i j}\right]$, where $b _{i j}=2^{i+j} a _{i j}$ for $1 \leq i, j \leq 3$. If the determinant of $P$ is 2 , then the determinant of the matrix $Q$ is
======= ####15. Let $P=[a _{i j}]$ be a $3 \times 3$ matrix and let $Q=[b _{i j}]$, where $b _{i j}=2^{i+j} a _{i j}$ for $1 \leq i, j \leq 3$. If the determinant of $P$ is 2 , then the determinant of the matrix $Q$ is
3e0f7ab6f6a50373c3f2dbda6ca2533482a77bed
(a) $2^{10}$
(b) $2^{11}$
(c) $2^{12}$
(d) $2^{13}$
(2012)
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Answer:
Correct Answer: 15. (d)
Solution:
- PLAN It is a simple question on scalar multiplication, i.e.
$\begin{vmatrix}k a _1 & k a _2 & k a _3 \\ b _1 & b _2 & b _3 \\ c _1 & c _2 & c _3\end{vmatrix}=k\begin{vmatrix}a _1 & a _2 & a _3 \\ b _1 & b _2 & b _3 \\ c _1 & c _2 & c _3\end{vmatrix}$
Description of Situation Construction of matrix,
$ \text { i.e. if } a=[a _{i j}] _{3 \times 3}$
= $\begin{vmatrix} a _{11} & a _{12} & a _{13} \\ a _{21} & a _{22} & a _{23} \\ a _{31} & a _{32} & a _{33} \end{vmatrix} $
Here, $\quad P=[a _{i j}] _{3 \times 3}$
$=\begin{vmatrix}a _{11} & a _{12} & a _{13} \\ a _{21} & a _{22} & a _{23} \\ a _{31} & a _{32} & a _{33}\end{vmatrix}$
$ Q=[b _{i j}] _{3 \times 3}$
$=\begin{vmatrix} b _{11} & b _{12} & b _{13} \\ b _{21} & b _{22} & b _{23} \\ b _{31} & b _{32} & b _{33} \end{vmatrix} $
where, $b _{i j}=2^{i+j} a _{i j}$
$ \begin{aligned} \therefore \quad|Q| & =\begin{vmatrix} 4 a _{11} & 8 a _{12} & 16 a _{13} \\ 8 a _{21} & 16 a _{22} & 32 a _{23} \\ 16 a _{31} & 32 a _{32} & 64 a _{33} \end{vmatrix} \\ & =4 \times 8 \times 16\begin{vmatrix} a _{11} & a _{12} & a _{13} \\ 2 a _{21} & 2 a _{22} & 2 a _{23} \\ 4 a _{31} & 4 a _{32} & 4 a _{33} \end{vmatrix} \\ & =2^{9} \times 2 \times 4\begin{vmatrix} a _{11} & a _{12} & a _{13} \\ a _{21} & a _{22} & a _{23} \\ a _{31} & a _{32} & a _{33} \end{vmatrix} \\ & =2^{12} \cdot|P|=2^{12} \cdot 2=2^{13} \end{aligned} $
