Question

Let $P=[a_{ij}]$ be a $3\times3$ matrix and let $Q=[b_{ij}]$ 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

Answer: (D)

Plan It is a simple question on scalar multiplication, i.e
$\begin{vmatrix}ka_1&ka_2&ka_3\\b_1&b_2&b_3\\c_1&c_2&c_3\end{vmatrix}= k\begin{vmatrix}a_1&a_2&a_3\\b_1&b_2&b_3\\c_1&c_2&c_3\end{vmatrix}$
Description of Situation Construction of matrix,
i.e. if $a=\left[a_{i j}\right]_{3 \times 3}=\begin{bmatrix}a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33}\end{bmatrix}$
Here, $ P=\left[a_{i j}\right]_{3 \times 3}=\begin{bmatrix}a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33}\end{bmatrix}$
$Q =\left[b_{i j}\right]_{3 \times 3}=\begin{bmatrix}b_{11} & b_{12} & b_{13} \\b_{21} & b_{22} & b_{23} \\b_{31} & b_{32} & b_{33}\end{bmatrix}$
where, $b_{i j}=2^{i+j} a_{i j}$
$\therefore |Q|=\begin{bmatrix}4 a_{11} & 8 a_{12} & 16 a_{13} \\8 a_{21} & 16 a_{22} & 32 a_{23} \\16 a_{31} & 32 a_{32} & 64 a_{33}\end{bmatrix}$
$=4 \times 8 \times 16 \begin{bmatrix}a_{11} & a_{12} & a_{13} \\ 2 a_{21} & 2 a_{22} & 2 a_{23} \\ 4 a_{31} & 4 a_{32} & 4 a_{33}\end{bmatrix}$
$=2^{9} \times 2 \times 4\begin{bmatrix}a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33}\end{bmatrix}=2^{12} \cdot|P|=2^{12} \cdot 2=2^{1,3}$