Question

let $A={\left[a_{ij}\right]}_{3\times 3}$ and $B={\left[a_{ij}\right]}_{3\times 3}$, where $b_{ij}=2^{i+j}{a}_{ij}$ for all $i,j$. if $i,j.$ If $|A|=2$, then $|B|=$

• A. $2^{11}$
• B. $2^{12}$
• C. $2^{13}$
• D. $2^{14}$

Answer 1

Answer: (C)

$2^{13}$

Given $A=[a_{i_j}{]}_{3\times 3}$ and $B[b_{i_j}{]}_{3\times 3}$

Where $b_{i_j}=2^{i+j}{a}_{i_j}$ and $|A|=2$

So,

$|B|=\left|\begin{array}{ccc}{2}^2{9}_{11}& 2^3{9}_{12}& 2^4{9}_{13}\\ 2^3{9}_{21}& 2^4{9}_{22}& 2^5{9}_{23}\\ 2^4{9}_{31}& 2^5{9}_{32}& 2^{6}933\end{array}\right|$

$|B|=2^2\times 2^3\times 2^4\left|\begin{array}{ccc}{9}_{11}& {29}_{12}& 2^2{9}_{12}\\ 9_{21}& {29}_{22}& 2^2{9}_{23}\\ 9_{31}& {29}_{32}& 2^2{9}_{33}\end{array}\right|$

$|B|=2^9\times 2\times 2\left|\begin{array}{ccc}{9}_{11}& 9_{12}& 9_{13}\\ 9_{21}& 9_{22}& 9_{23}\\ 9_{31}& 9_{32}& 9_{33}\end{array}\right|$

$\Rightarrow |B|=2^{12}\times 2$

$|B|=2^{13}$