Question

Let $P=\left[a_{ij}\right]$ be a $3\times 3$ matrix and let $Q=\left(b_{ij}\right)$ 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)

$2^{13}$

$P=(a_{ij}{)}_{3\times 3},b_{ij}=2^{i+j}\cdot a_{ij}$, $Q=(b_{ij}{)}_{3\times 3}$
$P=\left[\begin{array}{ccc}{a}_{11}& a_{12}& a_{13}\\ a_{21}& a_{22}& a_{23}\\ a_{31}& a_{32}& a_{33}\end{array}\right]\cdot |P|=2$
$Q=\left[\begin{array}{ccc}{b}_{11}& b_{12}& b_{13}\\ b_{21}& b_{22}& b_{23}\\ b_{31}& b_{32}& b_{33}\end{array}\right]=\left[\begin{array}{ccc}4a_{11}& 8a_{12}& 16a_{13}\\ 8a_{21}& 16a_{22}& 32a_{23}\\ 16a_{31}& 32a_{32}& 64a_{33}\end{array}\right]$
$=4\times 8\times 16\left|\begin{array}{ccc}{a}_{11}& a_{12}& a_{13}\\ 2a_{21}& 2a_{22}& 2a_{23}\\ 4a_{31}& 4a_{32}& 4a_{33}\end{array}\right|$
$=4\times 8\times 16\times 2\times 4\left|\begin{array}{ccc}{a}_{11}& a_{12}& a_{13}\\ a_{21}& a_{22}& a_{23}\\ a_{31}& a_{32}& a_{33}\end{array}\right|$
$=2^2\cdot 2^3\cdot 2^4\cdot 2^1\cdot 2^2\cdot 2^1=2^{13}$.