Finding basis of 3x3 matrix space

[black_hole]

Homework Statement

For my homework assignment, I'm supposed to find a basis for the space of 3x3 matrices that have zero row sums and separately for zero row columns. I am having a hard time with this as it seems to me that there are a lot of combinations I have to consider. For the first, it seems like the rows would have to consist of one 0, one 1, and one -1 in different orders... Is there a better way to do this other than brute force?

Homework Equations

The Attempt at a Solution

 

[HallsofIvy]

Well, a general 3 by 3 matrix can be written
$$\begin{bmatrix}a & b & c \\ d & e & f \\ g & h & i\end{bmatrix}$$.
The condition that "row sums are 0" means that a+ b+ c= 0, d+ e+ f= 0, and g+ h+ i= 0. Solve that for, say, a, d, and e and replace them in the matrix. Since the space of all 3 by 3 matrices is 9 dimensional, and you have 3 "conditions", you will want 6 basis matrices.