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)

$2^{13}$

$P\left(X\right)=\left|\begin{array}{ccc}{a}_{11}& a_{12}& a_{13}\\ a_{21}& a_{22}& a_{23}\\ a_{31}& a_{32}& a_{33}\end{array}\right|$

$Q=\left[bij\right]=2^{i+j}{a}_{ij}$
$Q=\left|\begin{array}{ccc}{2}^2{a}_{11}& 2^3{a}_{12}& 2^4{a}_{13}\\ 2^3{a}_{21}& 2^4{a}_{22}& 2^5{a}_{23}\\ 2^4{a}_{31}& 2^5{a}_{32}& 2^6{a}_{33}\end{array}\right|$
Taking $2^2,2^3,and{2}^4$ common from 1st, 2nd and 3rd row respectively, we get
$Q=2^9\left|\begin{array}{ccc}{a}_{11}& 2a_{12}& 2^2{a}_{13}\\ a_{21}& 2a_{22}& 2^2{a}_{23}\\ a_{31}& 2a_{32}& 2^2{a}_{33}\end{array}\right|$

Now taking $2,2^2$ common from 2nd and 3rd column respectively, we get

$Q=2^9\times 2^3\left|\begin{array}{ccc}{a}_{11}& a_{12}& a_{13}\\ a_{21}& a_{22}& a_{23}\\ a_{31}& a_{32}& a_{33}\end{array}\right|$
$Q=2^9\times 2^3\mid P\mid =2^9\times 2^4$ ($\because |P|=2$)

$\therefore |Q|=2^{13}$