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Matrix Inverse Calculator

Invert any square matrix up to 6×6 using Gauss-Jordan elimination on [A|I]. Detects singular matrices and verifies the result with A × A⁻¹ = I.

Inverse A⁻¹ (2×2)

0.6-0.7
-0.20.4

Verification

A × A⁻¹ was multiplied back out: the largest deviation from the identity matrix is 2.22e-16 — effectively zero, so the inverse checks out. Method: Gauss-Jordan elimination on the augmented matrix [A | I] with partial pivoting.

What is the Matrix Inverse Calculator?

The ByteTools Matrix Inverse Calculator computes the inverse of any square matrix up to 6×6. It augments your matrix with the identity matrix and runs Gauss-Jordan elimination with partial pivoting on [A|I]; when the left side becomes the identity, the right side is exactly A⁻¹ — the textbook method, executed reliably in floating point.

  • Inverts square matrices up to 6×6
  • Gauss-Jordan elimination on the augmented [A|I] matrix
  • Partial pivoting for numerical stability
  • Clear singular-matrix detection with tolerance
  • Built-in A × A⁻¹ verification against the identity
  • 100% private — runs entirely in your browser

How to use the Matrix Inverse Calculator

  1. 1

    Type the square matrix: one row per line, values separated by spaces or commas.

  2. 2

    The tool checks that the matrix is square (up to 6×6).

  3. 3

    Read the inverse matrix, or the "singular — no inverse" message.

  4. 4

    Check the verification row showing A × A⁻¹ ≈ I.

  5. 5

    Copy the inverse with one click.

About the Matrix Inverse Calculator

The ByteTools Matrix Inverse Calculator computes the inverse of any square matrix up to 6×6. It augments your matrix with the identity matrix and runs Gauss-Jordan elimination with partial pivoting on [A|I]; when the left side becomes the identity, the right side is exactly A⁻¹ — the textbook method, executed reliably in floating point.

If a pivot falls below a small numerical tolerance the matrix is singular, and the tool says so plainly: a matrix with determinant zero has no inverse. When an inverse does exist, the calculator also multiplies A by the computed A⁻¹ and shows how close the product is to the identity, so you can trust the result.

Everything runs 100% locally in your browser — no uploads, no accounts, no server. Type the matrix, read the inverse, copy it out.

Frequently asked questions

How do you find the inverse of a matrix?

Write the matrix side by side with the identity matrix as [A|I], then apply row operations until the left half becomes the identity. The right half is then A⁻¹. For a 2×2 matrix [[a,b],[c,d]] there is also the shortcut: swap a and d, negate b and c, and divide everything by the determinant ad − bc.

Which matrices have no inverse?

A matrix has no inverse exactly when its determinant is zero — such matrices are called singular. That happens when rows or columns are linearly dependent, for instance when one row is a multiple of another. This calculator detects the case and reports it instead of returning garbage.

What is the inverse of a 2x2 matrix?

For [[a,b],[c,d]] with determinant D = ad − bc ≠ 0, the inverse is (1/D) × [[d,−b],[−c,a]]. For example [[4,7],[2,6]] has D = 10, so the inverse is [[0.6,−0.7],[−0.2,0.4]]. You can verify this in the tool and see A × A⁻¹ come out as the identity.

Why does my nearly-singular matrix give huge numbers?

When the determinant is very close to zero, the true inverse genuinely contains very large entries, and tiny rounding errors get amplified. The calculator uses partial pivoting to keep the arithmetic as stable as possible and flags matrices whose pivots underflow the singularity tolerance.

What is a matrix inverse used for?

Inverses solve linear systems (x = A⁻¹b), undo linear transformations, and appear throughout statistics, computer graphics and engineering. In practice large systems are solved by elimination rather than explicitly inverting, but the inverse itself is fundamental for theory and for small matrices.

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