Matrices that represent rotations So the question is
What 3 by 3 matrices represent the transformations that
a) rotate the x-y plane, then x-z, then y-z through 90?
I believe this is the matrix that rotates the xy plane 
\begin{bmatrix}
0 &-1 &0 \\
1 &0 &0 \\
0 &0 &1 \\
\end{bmatrix}
But I couldn't think of a rotation that rotates the xz axis. Okay, I'm going to make y axis stay where it is, and rotate the $xz$ plane around it, but still which way should I rotate it? I have no idea. There are two ways to rotate it one in which z axis comes to the posisition of the axis where as x axis becomes -z, the other one is x axis becoming the z axis and z axis becoming -x. Which one to choose? and Why?
 A: Hint:
since you are a beginner I give you a general, simple and powerful method that works well in many situations.
You can interpret the action of your matrix
\begin{bmatrix}
0 &-1 &0 \\
1 &0 &0 \\
0 &0 &1 \\
\end{bmatrix}
You can interpret the action of your matrix looking at the columns. The first column is the transformation of the vector $[1,0,0]^T$, the second column is the transformation of the vector $[0,1,0]^T$ and the third columns is the transformation of $[0,0,1]^T$ (you can easily test this, that is a general result, true for all matrices). 
These three vectors are the standard basis of the vector space, and any vector can be expressed as a linear combination of them, and the transformed vector is the same linear combination of the transformed vectors of the basis (this is what ''linear transformation'' means).
Now you can see that your matrix does not change the component of a vector in the direction of the $z$ axis, but change a componet in the direction of the $x$axis ( the direction of $[1,0,0]^T$)  to $[0,1,0]^T$ and a component in the direction of the $y$ axis ($[0,1,0]^T$) to $[-1,0,0]^T$.
A simple vision of this transformation shows that it is a rotation around the $z$ axis ( the fixed points of the rotation) of $90°$ counterclockwise, as you have found.
Now you can use the same reasoning to find a rotation around the $y$ axis. If the angle of rotation is $90°$ counterclockwise (this is the usual convention) than you see that the $x$ axis transforms as:
$$
\begin{bmatrix}
1\\0\\0
\end{bmatrix}
\quad \rightarrow \quad
\begin{bmatrix}
0\\0\\-1
\end{bmatrix}
$$
and the $z$ axis transforms as:
$$
\begin{bmatrix}
0\\0\\1
\end{bmatrix}
\quad \rightarrow \quad
\begin{bmatrix}
1\\0\\0
\end{bmatrix}
$$
and, since the $y$ axis does not change, the matrix is:
$$
\begin{bmatrix}
0&0&-1\\
0&1&0\\
1&0&0
\end{bmatrix}
$$
Now you can use this method to find the other matrices in your question.
Note that your doubt: ''the other one is x axis becoming the z axis and z axis becoming -x'', is simply the clockwise rotation.
