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Find the Matrix Inverse

Enter square matrix dimensions, generate input matrix $A$, fill values, and calculate inverse using augmented matrix $[A | I]$.
Matrix entries can be real or complex, e.g. 3+2i, -1/2-i, 0.2i, etc.
(Matrix must be square)

For a square matrix $A$, the inverse $A^{-1}$ satisfies:

$ A \cdot A^{-1} = A^{-1} \cdot A = I $

Existence:

  • $A^{-1}$ exists if and only if $\det(A) \neq 0$ (matrix is invertible or non-singular)
  • If $\det(A) = 0$, the matrix is singular and has no inverse

Computing via Gauss-Jordan:

Form the augmented matrix $[A | I]$ and row-reduce to $[I | A^{-1}]$

Properties:

  • $(A^{-1})^{-1} = A$
  • $(AB)^{-1} = B^{-1}A^{-1}$
  • $(A^T)^{-1} = (A^{-1})^T$
Matrix size:

Enter the matrix size and click Generate Matrix to create an n×n input grid.

Display Values as:
1 — Compute
Computes the inverse via augmented row reduction [A | I] → [I | A⁻¹].
Resets all dimensions, matrix entries, and results.
Cite this tool
DOI: https://doi.org/10.5281/zenodo.20981241
Kapita, S. (2026). Find the Matrix Inverse. Math Tools. https://doi.org/10.5281/zenodo.20981241