Let `A=[a_("ij")]_(3xx3), B=[b_("ij")]_(3xx3)` and `C=[c_("ij")]_(3xx3)` be any three matrices, where `b_("ij")=3^(i-j) a_("ij")` and `c_("ij")=4^(i-j) b_("ij")`. If det. `A=2`, then det. `(BC)` is equal to _______ .

Answer

4

Explanation

det. `B=|(a_(11),a_(12)/3,a_(13)/3^(2)),(3a_(21),a_(22),a_(23)/3),(9a_(32),3a_(32),a_(33))|=3xx9|(a_(11),a_(12)/3,a_(13)/3^(2)),(a_(21),a_(22)/3,a_(23)/9),(a_(32),a_(32)/3,a_(33)/9)|`
(Dividing `C_(2)` by 3 and `C_(3)` by 9)
`=|(a_(11),a_(12),a_(13)),(a_(21),a_(22),a_(23)),(a_(32),a_(32),a_(33))|`
`="det. "A=2`
Similarly, det, `C=` det. `A=2`
`:.` det. `(BC)=` (det. B) (det. C) `=2xx2=4`