Question

A set of homogeneous simultaneous equations is given by
x + ky = 0
kx + 3y = 0
Calculate the two values of k that lead to non-trivial solutions to these
equations and express y in terms of x for the two values.

I thought that for the solutions to be no trivial the determinant of the coefficients had to equal zero. From this I got k=+/- sqrt 3 but I am not sure of this is right.

Answer 1

Rewrite as
y = (-1/k)x and
y = (-k/3)x
There are an infinite number of solutions if the lines coincide, which happens when

1/k = k/3
k = +/-sqrt3
In this case,
y = x/sqrt3 or -x/sqrt3

Otherwise, there are no solutions at all, other than x=y=0. That may be what they call the "trivial" solution. Nevertheless, it is still a solution.