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:

$\left|\begin{array}{ccc}a& 1& 4\\ 0& 2& 1\\ 1& 1& 3\end{array}\right|=0.$

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

$a\cdot \left|\begin{array}{cc}2& 1\\ 1& 3\end{array}\right|+1\cdot \left|\begin{array}{cc}1& 4\\ 2& 1\\ \end{array}\right|=0.$

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