If `alpha` is a non-real cube root of `-2`, then the value of `|(1,2 alpha,1),(alpha^(2),1,3 alpha^(2)),(2,2 alpha,1)|`, is

Options

A. `-11`
B. `-12`
C. `-13`
D. 0

Answer

C

Explanation

We have, `alpha = (-2)^(1//3) w or , alpha = (-2)^(1//3) w^(2)`
`:. alpha^(3) = -2`
Now, `|(1,2 alpha,1),(alpha^(2),1,3 alpha^(2)),(2,2 alpha,1)|`
`= |(1,2 alpha,1),(alpha^(2),1,3 alpha^(2)),(1,0,0)| " " ["Applying" R_(3) rarr R_(3) - R_(1)]`
`= 6alpha^(3) - 1` [On expanding along `R_(3)`]
`= -13 " " [ :' alpha^(3) = -2]`