Question

Let $P=[a_{ij}]$ be a $3\times 3$ 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
$\left|\begin{array}{ccc}k{a}_1& k{a}_2& k{a}_3\\ b_1& b_2& b_3\\ c_1& c_2& c_3\end{array}\right|=k\left|\begin{array}{ccc}{a}_1& a_2& a_3\\ b_1& b_2& b_3\\ c_1& c_2& c_3\end{array}\right|$
Description of Situation Construction of matrix,
i.e. if $a={\left[a_{ij}\right]}_{3\times 3}=\left[\begin{array}{ccc}{a}_{11}& a_{12}& a_{13}\\ a_{21}& a_{22}& a_{23}\\ a_{31}& a_{32}& a_{33}\end{array}\right]$
Here, $P={\left[a_{ij}\right]}_{3\times 3}=\left[\begin{array}{ccc}{a}_{11}& a_{12}& a_{13}\\ a_{21}& a_{22}& a_{23}\\ a_{31}& a_{32}& a_{33}\end{array}\right]$
$Q={\left[b_{ij}\right]}_{3\times 3}=\left[\begin{array}{ccc}{b}_{11}& b_{12}& b_{13}\\ b_{21}& b_{22}& b_{23}\\ b_{31}& b_{32}& b_{33}\end{array}\right]$
where, $b_{ij}=2^{i+j}{a}_{ij}$
$\therefore |Q|=\left[\begin{array}{ccc}4a_{11}& 8a_{12}& 16a_{13}\\ 8a_{21}& 16a_{22}& 32a_{23}\\ 16a_{31}& 32a_{32}& 64a_{33}\end{array}\right]$
$=4\times 8\times 16\left[\begin{array}{ccc}{a}_{11}& a_{12}& a_{13}\\ 2a_{21}& 2a_{22}& 2a_{23}\\ 4a_{31}& 4a_{32}& 4a_{33}\end{array}\right]$
$=2^9\times 2\times 4\left[\begin{array}{ccc}{a}_{11}& a_{12}& a_{13}\\ a_{21}& a_{22}& a_{23}\\ a_{31}& a_{32}& a_{33}\end{array}\right]=2^{12}\cdot |P|=2^{12}\cdot 2=2^{1,3}$