Two lines will not intersect if they are either PARALLEL, (by definition parallel means that they will not cross), or if they are exactly the same line.

Part B

Parallel

To explore what happens if I try and solve a parallel system with matrices I will need a set of parallel lines equations.

These equations are parallel because they share the same gradient (m=2). This is what they look like.

We will need to rearrange the equations so that the constant term is on the right-hand side. So,

And then interpreting this using matrices we get

(coefficient matrix) (variable matrix) = (constant term matrix)

We can isolate matrix X, by pre-multiplying both sides by the inverse of

This means we now need to find this inverse. We already know how to find the inverses, recall that

So in our case we have

Look at the determinant of the matrix. (Remember the determinant is the ad-bc part).

The determinant here is 0. So this matrix is singular and no inverse exists.

That's what happens when we try and solve parallel lines using matrices. There is no solution because the matrix will have no inverse. This doesn't mean that every time you have no inverse the lines are parallel - but it could be one of the reasons.

Same Line

The other option is that the lines we are trying to solve are actually one line. This may seem a bit ridiculous, but even lines that are identical can have equations that look different, and maybe you don't notice this straight away.

Consider these lines,