Hi, I understand what these two ideas mean, but I'm a bit lost on how to actually sit down and work out algebraically whether a set contains both properties or not.

An example to illustrate my point:

Determine whether the set of all 2x2 matrices whose determinant is 0 is a subspace of M22 (all 2x2 matrices).

My thinking went something like this

Matrix 1. AD-BC

Matrix 2. FG-HJ

hence if (A+F)(D+G)-(B+H)(C+J)=0 then the subset was close under vector addition. What is wrong with my thought train here?

For multiplication I figured something like

h(AD-BC)=0

AD-BC=0/h

AD-BC=0

hence subset was closed under scalar multiplication, is this correct?

Thanks a heap in advance,

Nic