Question

For what values of α are vectors (1,1,2,1), (2,1,2,3), (1,4,2,1) (-1,3,5,α) are linearly inde-

pendent

Answer 1

Let $v_1=(1, 1, 2, 1),\ \ v_2=(2, 1, 2, 3),\ \ v_3=(1,4,2,1),\ \ v_4=(-1,3,5,\alpha).$

These vectors are linearly independent if and only if $\begin{vmatrix} 1&1&2&1\\2&1&2&3\\1&4&2&1\\-1&3&5&\alpha \end{vmatrix}\neq0$ .

$\begin{vmatrix} 1&1&2&1\\2&1&2&3\\1&4&2&1\\-1&3&5&\alpha \end{vmatrix}=\begin{vmatrix}1&1&2&1\\0&-1&-2&1\\0&3&0&0\\0&4&7&\alpha+1 \end{vmatrix}= \begin{vmatrix} 1&1&2&1\\0&-1&-3&1\\0&0&-6&3\\0&0&-1&\alpha+5 \end{vmatrix}=-\begin{vmatrix} 1&1&2&1\\0&-1&-3&1\\0&0&-1&\alpha+5 \\0&0&-6&3 \end{vmatrix} =-\begin{vmatrix} 1&1&2&1\\0&-1&-3&1\\0&0&-1&\alpha+5 \\0&0&0&-6\alpha-27 \end{vmatrix}=-1\cdot(-1)\cdot(-1)\cdot(-6\alpha-27)=6\alpha+27\neq 0$

Answer: for all values $\alpha\neq -4.5$