Let `P=[a_(i j)]` be a `3xx3` matrix and let `Q=[b_(i j)],w h e r eb_(i j)=2^(i+j)a_(i j) for 1lt=i ,jlt=3.` If the determinant of `P` is 2, then the determinant of the matrix `Q` is

Options

A. `2^(11)`
B. `2^(12)`
C. `2^(13)`
D. `2^(10)`

Answer

C

Explanation

We have, `abs(Q) = abs((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)))`
` =2^(2) cdot 2^(3)cdot2^(4) abs(( a_(11) ,a_(12), a_(13)),(2a_(21),2a_(22),2 a_(23) ),(2^(2)a_(31),2^(2)a_(32),2^(2)a_(33)))`
` =2^(9) cdot 2cdot2^(2) abs(( a_(11) ,a_(12), a_(13)),(a_(21),a_(22), a_(23) ),(a_(31),a_(32),a_(33))) = 2^(12) abs(P)`
`therefore abs(Q)=2^(12)xx2=2^(13)`