Proving Linear Dependence of 3 Vectors with 3 Unknowns

[lo2]

I have these 3 vectors. And I have to prove that they are dependent.

a=(-1,7,-4)

b=(3,5,-6)

a=(15,-1,-12)

I have tried to make 3 equtations with 3 unknows and the solve it. But I just get that the constants should be 0, what they should not be in order to prove they are dependent?

 

[courtrigrad]

So $$A = \begin{bmatrix} -1&7&4 \\ \end{bmatrix}$$

So $$B = \begin{bmatrix} 3&5&-6 \\ \end{bmatrix}$$

So $$C = \begin{bmatrix} 15&-1&-12 \\ \end{bmatrix}$$

For the vectors to be linearly dependent, $$\left| \begin{array} {ccc} -1 & 7 & 4 \\ 3 & 5& -6 \\ 15 & -1 & -12 \\ \end{array} \right|= 0$$

 

[Hootenanny]

Alternatively, if you don't like evaluating determinants you could try row reduction to the echelon form (obtaining a final row of zeros would indicate LD).