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

`|Q|=|(2^(2)a_(11),2^(3)a_(12),2^(2)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). 2^(3). 2^(4)|(a_(11),a_(12),a_(13)),(2a_(21),2a_(22),2a_(23)),(2^(2)a_(31),2^(2)a_(32),2^(2) a_(33))|`
`=2^(9). 2. 2^(2) |(a_(11),a_(12),a_(13)),(a_(21),a_(22),a_(23)),(a_(31),a_(32),a_(33))|`
`=2^(12) |P|`
`=2^(13)`