Let `P=[a_("ij")]` be a `3xx3` matrix and let `Q=[b_("ij")]`, 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

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)`