Inverse Matrix in Python My understanding is that I can use Python to initialize my matrix and then apply an inverse function to find the solution.
You can use matlab but:

1.) you will get a numerical approximation of the inverse,

2.) you won't learn anything from that.

Rather, try doing what the question asks you - work out what $z_1, z_2, z_3$ are in terms of $b_1, b_2, b_3$ (by writing out the equations and rewriting them a bit) or use any other technique to calculate an inverse of $3\times3$ matrix.
I'm not sure I understand your notation: 


  deltaz = [-1 1 0][z1] = [z2 - z1] = [b1] = b [0 -1 1] [z2] [z3 - z2] [b2] [0 0 -1] [z3] [0 - z3] [b3]


Are you attempting Python pseudocode or are you defining a matrix? If the latter, please use MathJax for a more readable format.