For which real values of $\lambda$ do the following vectors form a linearly dependent set in $R^3$?
$v_1 = \langle \lambda, -1/2, -1/2 \rangle$,$v_2 = \langle -1/2, \lambda, -1/2 \rangle$,$v_3 = \langle -1/2, -1/2, \lambda \rangle$
Could I simply put $v_1,v_2,v_3$ into a 3 by 3 matrix and find the determinant such that it is equal to 0?
Best Answer
Your idea is exactly right.
Let $$ A= \left[\begin{array}{rrr} \lambda & -\frac{1}{2} & -\frac{1}{2} \\ -\frac{1}{2} & \lambda & -\frac{1}{2} \\ -\frac{1}{2} & -\frac{1}{2} & \lambda \end{array}\right] $$ Then $$ \det A =\frac{1}{4} \, {\left(4 \, \lambda^{2} - 1\right)} \lambda - \frac{1}{2} \, \lambda - \frac{1}{4} =\frac{1}{4} \, {\left(2 \, \lambda + 1\right)}^{2} {\left(\lambda - 1\right)} $$ This proves that your vectors are linearly independent if and only if $\lambda\neq-1/2$ and $\lambda\neq 1$.