A 4x4 matrix

Recently in one of my tutorial classes we came across an interesting problem. At first glance it was merely a system of four equations with four variables which we had to solve by means of a matrix. So we (or rather the students) went about carrying out the row operations, reducing the matrix to row echolen form.
(Note that this was a first year class and the concept was still very new to them, hence many struggled to complete these operations.)

This is an example of a 3x3 matrix in row echelon form:

Now, solving this system we find that it has infinitely many solutions, and the solutions turns out to be the equation of a plane.

I'll give you a moment to consider this...

We have four objects in a four dimensional space which intersects in a three dimensional object, a plane.
What does these 4D objects look like? Even though don't know, we are able to show how they would intersect.

Interesting? I thought so.

Happy 4D thinking!