Question

Let $P=[a_{ij}]$ be a $3\times3$ matrix and let $Q=[b_{ij}]$ where $b_{ij}=2^{i+j} a_{ij}$ for $1 \le i, j \le$.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)

$2^{13}$

Answer 2

Plan It is a simple question on scalar multiplication, i.e
\begin{array}|ka_1&ka_2&ka_3\\\b_1&b_2&b_3\\\c_1&c_2&c_3\\\\\end{array}=\begin{array}|a_1&a_2&a_3\\\b_1&b_2&b_3\\\c_1&c_2&c_3\\\\\end{array}

Description of Situation Construction of matrix,
I.e. if $a=[a_{ij}]_{3\times3}\begin{array}|a_{11}&a_{12}&a_{13}\\\a_{21}&a_{22}&a_{23}\\\a_{31}&a_{32}&a_{33}\end{array}$

Here, $P[a_{ij}]_{3\times3}\begin{array}|a_{11}&a_{12}&a_{13}\\\a_{21}&a_{22}&a_{23}\\\a_{31}&a_{32}&a_{33}\\\\\end{array}$

$Q[b_{ij}]_{3\times3}\begin{array}|b_{11}&b_{12}&b_{13}\\\b_{21}&b_{22}&b_{23}\\\b_{31}&b_{32}&b_{33}\\\\\end{array}$

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

$\therefore | Q |=\begin{array}|4 a_{11}&8 a_{12}&16 a_{13}\\\8 a_{21}&16 a_{22}&32 a_{23}\\\16 a_{31}&32 a_{32}&64 a_{33}\\\\\end{array}$

$=4\times8\times16=\begin{array}| a_{11}&a_{12}&a_{13}\\\2 a_{21}&2 a_{22}&2 a_{23}\\\4 a_{31}&4 a_{32}&4 a_{33}\\\\\end{array}$

$=2^9\times 2\times 4=\begin{array}| a_{11}&a_{12}&a_{13}\\\a_{21}&a_{22}&a_{23}\\\a_{31}&a_{32}&a_{33}\\\\\end{array}=2^{12}.2=2^{13}$