Prove that the value of each the following determinants zero: `(a)|{:(a_(1),,la_(1)+mb_(1),,b_(1)),(a_(2),,la_(2)+mb_(2),,b_(2)),( a_(3),,la_(3)+mb_(3),,b_(3)):}|`

Answer

-1

Explanation

(a) Operating `C_(2)-lC_(1)-mC_(3).` we get the value of the determinant is 0
(b) Operating `C_(1) to C_(1) +C_(2)+C_(3)` we get the value of the determinant is 0
(C ) `Delta=|{:(log x,,log y,,log z),(log 2+log x,,log 2+logy,,log 2+log z),(log 3+logx,,log 3+log y,,log 3+logz):}|`
Operating `R_(3) to R_(3)-R_(1) " and " R_(2) to R_(2) -R_(1)` we get
`Delta =|{:(log x,,log y,,log z),(log 2,,log 2,,log 2),(log 3,,log 3,,log 3):}|`
`=0" "(:'R_(2) " and "R_(3) " are same ")`
(d) Using `C_(1) to C_(1)-C_(2)-4C_(3)` we have
`|{:(0,,(a^(x)-a^(-x))^(2),,1),(0,,(b^(y)-b^(-y))^(2),,1),(1,,(c^(z)-c^(-z))^(2),,1):}|=0`
(e ) Applying `C_(3)toC_(3)-C_(1)` we get
`|{:(sin^(2)(x+(3pi)/(2)),,sin^(2)(x+(5pi)/(2)),,sin(2x+5pi)sin(2pi)),(sin(x+(3pi)/(2)),,sin(x+(5pi)/(2)),,2cos (x+(5pi)/(2))sin(pi)),(sin(x-(3pi)/(2)),,sin(x-(5pi)/(2)),,2cos (x-(5pi)/(2))sin(pi)):}|=0`
`[:. ` all elements of `C_(3) ` are zero]