[Python] How to solve a 3D system of equations in Python

I have a system of equations like this:



The problem is that every component aij, xij and bi are vectors, or lists (change over years).

So we have a nxnxd numpy matrix with the vector A (nxnxd), B (nxd) and X (nxnxd) and we need to solve that system. Is that possible in a vectorized way with numpy?

We know that the system always have a solution This is a reduced example:

import pandas as pd
import sympy as sp

# This code snippet demonstrates how to create a DataFrame with symbolic equations
# Constant nodes are vectors (every entrance is the value per year)
cte_nodes = {"a_11": [1, 2, 3], "a_12": [4, 5, 6], "a_13": [7, 8, 9],
"a_21": [10, 11, 12], "a_22": [13, 14, 15], "a_23": [16, 17, 18],
"a_31": [19, 20, 21], "a_32": [22, 23, 24], "a_33": [25, 26, 27]}
# Every row is a system of equations, and every column is a variable x1, x2, x3
# The equations are represented as a DataFrame
eqs_df = pd.DataFrame([[sp.sympify("x1/a11 + a12"), sp.sympify("x1/a11 + a12"), sp.sympify("x1/a11 + a12")],
[sp.sympify("x2/a21 + a22"), sp.sympify("x2/a21 + a22"), sp.sympify("x1/a11 + a12")],
[sp.sympify("x3/a31 + a32"), sp.sympify("x3/a31 + a32"), sp.sympify("x1/a11 + a12")]]).T
eqs_df.columns = ["x1", "x2", "x3"]

# The system always have solution


Again, the code below is just a simplification, but speed matters, because the original matrices are much larger. Thanks in advance.

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