Subspace Problems: Which R^n*n Subsets are Subspaces?

[dkyo]

Homework Statement

Which of the following subsets of R^n*n are in fact subspaces of R^n*n

1) The symmetric matrices
2) The diagonal matrices
3) The nonsingular matrices
4) The singular matrices
5) The triangular matrices
6) The upper triangular matrices
7) All matrices that commute with a given matrix A
8) All matices such that A^2 = A
9) All matrices such that trace(A) = 0

Can anyone give me a detailed solution for this questions?

Homework Equations

The Attempt at a Solution

 

[nicktacik]

First of all, what are the qualities a subset needs to have to be a subspace of R^n*n?

 

[Dick]

No one will 'give you a detailed solution', they will only help you to arrive at your own solution. nicktacik has suggested how to start thinking about the problem. I suggest you start.