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)for1lt=i ,jlt=3.` If the determinant of `P` is 2, then the determinant of the matrix `Q` is `2^(10)` b. `2^(11)` c. `2^(12)` d. `2^(13)`

Options

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

Answer

D

Explanation

We have, `Q = [b_(ij)], " where " b_(ij) = 2^(i +j) a_(ij)`
`:. Q = |(b_(11),b_(12),b_(13)),(b_(21),b_(22),b_(23)),(b_(31),b_(32),b_(33))|`
`rArr Q = |(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))|`
`rArr Q = 2^(2) xx 2^(3) xx 2^(4) |(a_(11),2a_(12),2^(2) a_(13)),(a_(21),2a_(22),2^(2) a_(23)),(a_(31),2a_(32),2^(2) a_(33))|`
`rArr Q = 2^(9) xx 2 xx 2^(2) |(a_(11),a_(12),a_(13)),(a_(21),a_(22),a_(23)),(a_(31),a_(32),a_(33))| = 2^(12) P`
`rArr |Q| = 2^(12) |P| = 12^(12) xx 2 = 2^(13)`