Determine if a set is closed under scalar multiplication | Linear algebra I

In this video I went through an example from an intro linear algebra course dealing with closure under scalar multiplication. The problem was to determine if the set U of all 2x2 matrices with third entry equal to 0 is closed under scalar multiplication.

If something is closed, you can't get out. A set being 'closed under scalar multiplication' means that you can't get outside of that set just with multiplication by a constant. In this case, since multiplying a matrix by a constant won't change the fact that it has a zero in the third entry, this set U is closed under scalar multiplication.

Formally how we see if a set is closed under scalar multiplication is by taking some arbitrary element in the set, let's call it v, and see if the vector k times v is also in the set, for any real number (constant) k. If scaling by any constant isn't enough to get you out of the set, then the set is closed under scalar multiplication.