Science:Math Exam Resources/Courses/MATH152/April 2016/Question A 18/Solution 1

We write these three vectors as the column vectors of a matrix. The determinant of a matrix is zero if its column vectors are dependent:

${\begin{vmatrix}a&1&4\\0&2&1\\1&1&3\end{vmatrix}}=0.$

If we expand the first column to calculate the determinant, we have

$a\cdot {\begin{vmatrix}2&1\\1&3\end{vmatrix}}+1\cdot {\begin{vmatrix}1&4\\2&1\\\end{vmatrix}}=0.$

This implies $a(2\times 3-1\times 1)+1(1\times 1-2\times 4)=5a-7=0.$ Thus, $\color {blue}a={\frac {7}{5}}.$