find all values of a for which the system has nontrivial solutions

Question

So i was given this question.

In each of the following, find all values of a for which the system has nontrivial solutions, and determine all solutions in each case.

a) x - 2y + z = 0

x + ay - 3z = 0

-x + 6y - 5z = 0

Here is my attempt:

$$\left[\begin{array}{ccc|c} 1 & -2 & 1 & 0 \\ 1 & a & -3 & 0 \\ -1 & 6 & -5 & 0 \end{array}\right] \sim \left[\begin{array}{ccc|c} 1 & -2 & 1 & 0 \\ 0 & a+6 & -8 & 0 \\ -1 & 6 & -5 & 0 \end{array}\right]\\ \sim \left[\begin{array}{ccc|c} 1 & -2 & 1 & 0 \\ 0 & a+6 & -8 & 0 \\ 0 & 2 & -2 & 0 \end{array}\right]$$

I'm stuck on how to go about this, what really confuses me is the variable a, and how to subtract/add/multiply/divide when there is a variable between numbers.

Answer 1

You just need to go a little further with the row reduction.

$$\left[\begin{array}{ccc|c} 1 & -2 & 1 & 0 \\ 0 & a+6 & -8 & 0 \\ 0 & 2 & -2 & 0 \end{array}\right]\sim \left[\begin{array}{ccc|c} 1 & -2 & 1 & 0 \\ 0 & a+6 & -8 & 0 \\ 0 & 1 & -1 & 0 \end{array}\right]\sim \left[\begin{array}{ccc|c} 1 & -2 & 1 & 0 \\ 0 & a+6 & -8 & 0 \\ 0 & 0 & a-2 & 0 \end{array}\right].$$

So, in order for the system to have non-trivial solutions, we must have $a=2$. Then we have a row of zeros.