SATHEE: Matrices and Determinants 2 Question 13

Matrices and Determinants 2 Question 13

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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

======= ####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) $2^{11}$

(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.

$| \begin{matrix} k a_{1} & k a_{2} & k a_{3} \\ b_{1} & b_{2} & b_{3} \\ c_{1} & c_{2} & c_{3} \end{matrix} | = k | \begin{matrix} a_{1} & a_{2} & a_{3} \\ b_{1} & b_{2} & b_{3} \\ c_{1} & c_{2} & c_{3} \end{matrix} |$

Description of Situation Construction of matrix,

$i.e. if a = [a_{i j}]_{3 \times 3}$

= $| \begin{matrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{matrix} |$

Here, $P = [a_{i j}]_{3 \times 3}$

$= | \begin{matrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{matrix} |$

$Q = [b_{i j}]_{3 \times 3}$

$= | \begin{matrix} b_{11} & b_{12} & b_{13} \\ b_{21} & b_{22} & b_{23} \\ b_{31} & b_{32} & b_{33} \end{matrix} |$

where, $b_{i j} = 2^{i + j} a_{i j}$

$\begin{aligned} ∴ | Q | & = | \begin{array}{c} 4 a_{11} & 8 a_{12} & 16 a_{13} \\ 8 a_{21} & 16 a_{22} & 32 a_{23} \\ 16 a_{31} & 32 a_{32} & 64 a_{33} \end{array} | \\ = 4 \times 8 \times 16 | \begin{array}{c} a_{11} & a_{12} & a_{13} \\ 2 a_{21} & 2 a_{22} & 2 a_{23} \\ 4 a_{31} & 4 a_{32} & 4 a_{33} \end{array} | \\ = 2^{9} \times 2 \times 4 | \begin{array}{c} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{array} | \\ = 2^{12} \cdot | P | = 2^{12} \cdot 2 = 2^{13} \end{aligned}$