SOLVE

Computes the exact solution of the linear system Ax = b for the unknown x. The matrix A must be square and non-singular. The function supports various matrix types (general, symmetric, Hermitian, positive definite) which can be specified to use optimized solvers.

Excel Usage

=SOLVE(a_matrix, b_vector, assume_a)

a_matrix (list[list], required): Square 2D array of numeric coefficients.
b_vector (list[list], required): 2D array representing the right-hand side (can be a vector or matrix).
assume_a (str, optional, default: “gen”): Type of matrix A to assume (allows using optimized solvers).

Returns (list[list]): 2D array representing the solution x.

Example 1: Solve 2x2 system

Inputs:

a_matrix		b_vector
3	1	9
1	2	8

Excel formula:

=SOLVE({3,1;1,2}, {9;8})

Expected output:

Result
2
3

Example 2: Solve symmetric system assumption

Inputs:

a_matrix		b_vector	assume_a
4	1	1	sym
1	3	2

Excel formula:

=SOLVE({4,1;1,3}, {1;2}, "sym")

Expected output:

Result
0.0909091
0.636364

Example 3: Solve with positive definite assumption

Inputs:

a_matrix			b_vector	assume_a
2	-1	0	1	pos
-1	2	-1	0
0	-1	2	1

Excel formula:

=SOLVE({2,-1,0;-1,2,-1;0,-1,2}, {1;0;1}, "pos")

Expected output:

Result
1
1
1

Example 4: Solve with matrix right-hand side

Inputs:

a_matrix		b_vector
1	2	5	6
3	4	7	8

Excel formula:

=SOLVE({1,2;3,4}, {5,6;7,8})

Expected output:

Result
-3	-4
4	5

Python Code

import numpy as np
from scipy.linalg import solve as scipy_solve

def solve(a_matrix, b_vector, assume_a='gen'):
    """
    Solve a linear matrix equation, or system of linear scalar equations.

    See: https://docs.scipy.org/doc/scipy/reference/generated/scipy.linalg.solve.html

    This example function is provided as-is without any representation of accuracy.

    Args:
        a_matrix (list[list]): Square 2D array of numeric coefficients.
        b_vector (list[list]): 2D array representing the right-hand side (can be a vector or matrix).
        assume_a (str, optional): Type of matrix $A$ to assume (allows using optimized solvers). Valid options: General, Symmetric, Hermitian, Positive Definite. Default is 'gen'.

    Returns:
        list[list]: 2D array representing the solution x.
    """
    try:
        def to2d(x):
            return [[x]] if not isinstance(x, list) else x

        a_matrix = to2d(a_matrix)
        b_vector = to2d(b_vector)

        if not isinstance(a_matrix, list) or not a_matrix or not all(isinstance(row, list) for row in a_matrix):
            return "Error: a_matrix must be a non-empty 2D list"

        n = len(a_matrix)
        if any(len(row) != n for row in a_matrix):
            return "Error: a_matrix must be square (n x n)"

        try:
            a = np.array(a_matrix, dtype=float)
            b = np.array(b_vector, dtype=float)
        except (ValueError, TypeError):
            return "Error: matrix entries must contain numeric values"

        if not np.all(np.isfinite(a)) or not np.all(np.isfinite(b)):
            return "Error: Input must contain only finite numbers"

        try:
            x = scipy_solve(a, b, assume_a=assume_a.lower())
        except np.linalg.LinAlgError as e:
            return f"Error: Linear algebra solver failed: {str(e)}"
        except Exception as e:
            return f"Error: {str(e)}"

        return x.tolist()

    except Exception as e:
        return f"Error: {str(e)}"

Online Calculator

a_matrix *

Square 2D array of numeric coefficients.

b_vector *

2D array representing the right-hand side (can be a vector or matrix).

assume_a

Type of matrix $A$ to assume (allows using optimized solvers).