Gaussian Elimination

The problem, from first principles

05-matrix-inverse computes A⁻¹ and uses it to solve Ax = b, but never explains how an inverse (or a solution) is actually found by hand in the first place — it's presented as a formula to apply, not a procedure. Gaussian elimination is that procedure: a systematic way to reduce a system of equations, step by step, until the answer is directly readable off the result. It's also the exact mechanism behind 11-lu-decomposition and behind why 05-matrix-inverse's inverse fails to exist for some matrices.

From theory to code

Implement gaussian_eliminate(A, b), reducing the augmented system [A | b] to upper-triangular form via row operations, then back_substitute(U, c), solving the resulting triangular system for x. The signatures and docstrings are already in the editor.

Constraints

• A is a square n × n NumPy array; b is a length-n NumPy array.
• Assume no row swaps are needed (every pivot encountered is non-zero) — the same assumption 11-lu-decomposition makes.
• gaussian_eliminate returns the reduced augmented system as (U, c): U upper-triangular, c the correspondingly transformed right-hand side.
• back_substitute returns the solution vector x such that Ux = c.

Hints