Matrix Inverse Calculator

Compute the inverse of a square matrix via LU decomposition with singularity detection.

About Matrix Inverse Calculator

The inverse A⁻¹ of a square matrix A satisfies A·A⁻¹ = I. This calculator uses LU decomposition with partial pivoting for numerical stability. If the matrix is singular (determinant effectively zero given a tolerance of 1e-12), no inverse exists.

Formulas

Inverse Definition

A · A⁻¹ = A⁻¹ · A = I

A: Square matrix
I: Identity matrix

LU Method

Solve L·U·xᵢ = eᵢ for each column of A⁻¹

L: Lower triangular
U: Upper triangular
eᵢ: i-th standard basis vector

Worked example

With the default values below, Matrix Inverse Calculator returns A⁻¹ of 3 -1 -5 2. Change any input above and every figure here is recalculated in your browser.

Inputs used

Matrix A (rows separated by newlines or semicolons; values by commas or spaces): 2 1 5 3

Results

A⁻¹: 3 -1 -5 2
Determinant: 1.00000000

Step by step

1Matrix A= 2 1 5 3
2det(A)= 1
3A⁻¹ (via LU with partial pivoting)= 3 -1 -5 2

Assumptions and limits

The result is derived only from the single input listed above. Anything not among them is not part of the calculation.
In particular, this calculator does not take inflation, tax, fees or charges or compounding frequency into account.
Figures are rounded for display to 8 decimal places, so totals built from several rounded parts can differ slightly from the unrounded arithmetic.

How these calculators are built and tested is described on the Methodology page. If a figure looks wrong, please report it.

Frequently Asked Questions

When does a matrix have no inverse?

A matrix is singular (not invertible) when its determinant is zero, meaning its rows or columns are linearly dependent.

Why use LU instead of cofactor expansion?

LU decomposition is O(n³) and numerically stable with partial pivoting, while cofactor expansion is O(n!) and impractical for matrices larger than 4×4.