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For which real values of x do the following vectors form a linearly dependent set in R3?

v1= (x, -1/9, -1/9)
v2= (-1/9, x, -1/9)
v3= (-1/9, -1/9, x)

  • Math -

    The vectors form a linearly dependent set in ℜ3 if the determinant of the matrix formed by the three vectors is zero.

    Evaluate the determinant:
    |x,k,k|
    |k,x,k|
    |k,k,x|
    in terms of x and k, where k can eventually substituted for -1/9.
    Solve the resulting cubic equation for x in terms of k. It actually factors nicely into (x+?)(x-??)².

    In fact, you should be able to determine ?? by inspection.

    Post if you need more help.

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