1. ## Subspaces

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

A. The invertible 3 x 3 matrices
B. The 3 x 3 matrices with determinant 0
C. The symmetric 3 x 3 matrices
D. The diagonal 3 x 3 matrices
E. The 3 x 3 matrices in reduced row-echelon form
F. The 3 x 3 matrices with trace 0 (the trace of a matrix is the sum of its diagonal entries)

I am not even sure how to begin, do you still apply the closure properties, if so how? Thank you for any help!

2. you have to test the closure of vector addition and scalar multiplication

consider A. The invertible 3x3 matrices will not have the zero matrix since it is not invertible, so the space of 3x3 matrices will not be a vector space in its own right

consider D. Multiplying two diagonal matices will give you a diagonal matrix, adding together two diagonal matrices will give you a diagonal matrix.

The rest you can do in a similar manner, just go throught the properties of the respective subsets

3. For (B) note that $\displaystyle A= \begin{pmatrix} 1& 0 \\ 0 & 0\end{pmatrix}$ and $\displaystyle B= \begin{pmatrix}0 & 0 \\ 0 & 1\end{pmatrix}$ both have determinant 0 and so are in the set. What about their sum?