The value of the determinant `|[log_a(x/y), log_a(y/z), log_a(z/x)], [log_b (y/z), log_b (z/x), log_b (x/y)], [log_c (z/x), log_c (x/y), log_c (y/z)]|`

Options

A. 1
B. -1
C. 0
D. `(1)/(6) log _(a) xyz`

Answer

C

Explanation

`Delta = |{:(log_(a)((x)/(y)),,log_(a)((y)/(z)),,log_(a)((z)/(x))),(log_(a^(2))((y)/(z)),,log_(a^(2))((z)/(x)),,log_(a^(2))((x)/(y))),(log_(a^(3))((z)/(y)),,log_(a^(3))((x)/(y)),,log_(a^(3))((y)/(z))):}|`
`=(1)/(2) xx(1)/(3) |{:(log_(a)((x)/(y)),,log_(a)((y)/(z)),,log_(a)((z)/(x))),(log_(a)((y)/(z)),,log_(a)((z)/(x)),,log_(a)((x)/(y))),(log_(a)((z)/(y)),,log_(a)((x)/(y)),,log_(a)((y)/(z))):}|`
Now ,`log_(a)((x)/(y)) +log_(a)((y)/(z))+log_(a)((z)/(y))`
`=log_(a)((x)/(y))((y)/(z))((z)/(x))=log_(a)1=0`
So applying operation `R_(1) to R_(1)+R_(2)+R_(3) " on " Delta ` we get
`Delta =(1)/(2)xx(1)/(3) |{:(0,,0,,0),(log_(a).((y)/(z)),,log_(a).((z)/(x)),,log_(a).((x)/(y))),(log_(a).((z)/(x)),,log_(a).((x)/(y)),,log_(a).((y)/(z))):}|=0`