Which of the following subsets of R^(3x3) are subspaces of R^(3x3)

The 3x3 matrices whose entries are all integers.

The 3x3 matrices with all zeros in the third row.

The symmetric 3x3 matrix.

the 3x3 matrices with trace zero (the trace of the matrix is the sum of its diagonal entries)

The 3x3 matrices whose entries are all greater than or equal to 0

the 3x3 matrices with determinant 0

I could use an explanation as to why or why not. Or at least some counterexamples.

You just need to show closure of linear combinations, and that can be shown that both "vector" addition and scalar multiplication are closed.

all integers: No. An irrational multiple of a nozero matrix always has noninteger elements

3rd row zero: Yes. Elements are added and multiplied itemwise, and any linear combination of zeros is zero.

trace zero: Yes. By elements, if both A and B have zero trace then