if `x ne 0 , y ne 0 ,z ne 0 " and " |{:(1+x,,1,,1),(1+y,,1+2y,,1),(1+z,,1+z,,1+3z):}|=0` then
`x^(-1) +y^(-1) +z^(-1)` is equal to

Options

A. `-1`
B. `-2`
C. `-3`
D. none of these

Answer

C

Explanation

`|{:(1+x,,1,,1),(1+y,,1+2y,,1),(1+z,,1+z,,3+3z):}|`
`=xyz |{:(1+(1)/(x),,(1)/(x),,(1)/(x)),(1+(1)/(y),,2+(1)/(y),,(1)/(y)),(1+(1)/(z),,1+(1)/(z),,3+(1)/(z)):}|`
`=xyz (3+(1)/(x)+(1)/(y)+(1)/(z)) |{:(1+(1)/(x),,(1)/(x),,(1)/(x)),(1+(1)/(y),,2+(1)/(y),,(1)/(y)),(1+(1)/(z),,1+(1)/(z),,3+(1)/(z)):}|`
`=xyz (3+(1)/(x)+(1)/(y)+(1)/(z)) |{:(1,,0,,0),(1+(1)/(y),,1,,-1),(1+(1)/(z),,0,,2):}|`
`=2xyz (3+(1)/(x)+(1)/(y)+(1)/(z)) `
Hence the given equation gives `x^(-1) +y^(-1) +z^(-1) =-3`