INV

Computes the matrix A^{-1} such that AA^{-1} = I, where I is the identity matrix. The input matrix must be square and non-singular. For solving linear systems, using solve is generally more efficient and stable than computing an explicit inverse.

Excel Usage

=INV(matrix)

matrix (list[list], required): Square 2D array of numeric values to invert.

Returns (list[list]): 2D array representing the matrix inverse.

Example 1: Inverse of 2x2 matrix

Inputs:

matrix
1	2
3	4

Excel formula:

=INV({1,2;3,4})

Expected output:

Result
-2	1
1.5	-0.5

Example 2: Inverse of diagonal 3x3 matrix

Inputs:

matrix
2	0	0
0	4	0
0	0	0.5

Excel formula:

=INV({2,0,0;0,4,0;0,0,0.5})

Expected output:

Result
0.5	0	0
0	0.25	0
0	0	2

Example 3: Inverse of identity matrix

Inputs:

matrix
1	0
0	1

Excel formula:

=INV({1,0;0,1})

Expected output:

Result
1	0
0	1

Example 4: Inverse of matrix with negative entries

Inputs:

matrix
-2	1
5	3

Excel formula:

=INV({-2,1;5,3})

Expected output:

Result
-0.272727	0.0909091
0.454545	0.181818

Python Code

import numpy as np
from scipy.linalg import inv as scipy_inv

def inv(matrix):
    """
    Compute the inverse of a square matrix.

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

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

    Args:
        matrix (list[list]): Square 2D array of numeric values to invert.

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

        matrix = to2d(matrix)

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

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

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

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

        try:
            res = scipy_inv(a)
        except np.linalg.LinAlgError:
            return "Error: Matrix is singular and cannot be inverted"
        except Exception as e:
            return f"Error: {str(e)}"

        return res.tolist()

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

Online Calculator

matrix *

Square 2D array of numeric values to invert.