Matrices and Determinants 2 Question 13
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
(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, $\quad P=[a_{i j}]_{3 \times 3}$
$=\begin{vmatrix}a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33}\end{vmatrix}$
$ \mathrm{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} $
