Inverse Matrix Calculator

Find the inverse of a matrix using Gauss-Jordan elimination on the augmented matrix [A | I]. Every row operation is shown.

100% Free No Sign-Up Calculated in Your Browser

Matrix Size

Matrix A

Find the inverse of a 2×2, 3×3, or 4×4 matrix using Gauss-Jordan elimination on the augmented matrix [A | I], with every row operation shown.

How It Works

How Inverse Matrix Calculator Works

A matrix's inverse, written A⁻¹, is the matrix that satisfies A × A⁻¹ = I (the identity matrix) — multiplying a matrix by its inverse always produces the identity. Not every matrix has one: only "non-singular" square matrices, whose determinant is not zero, are invertible.

To find it, the calculator augments A with the identity matrix of the same size to form [A | I], then applies row operations — swapping rows for numerical stability, scaling a row so its pivot becomes 1, and subtracting multiples of one row from another to zero out the rest of that column. It repeats this column by column until the left half becomes the identity matrix.

Once the left side is I, whatever remains on the right side is exactly A⁻¹ — this works because every row operation applied to A is simultaneously applied to I, effectively "recording" the sequence of operations that undoes A. If a pivot column can't be made non-zero at any step, the matrix is singular and has no inverse.

Worked Example

See It In Action

For A = [[4,7],[2,6]]: det(A) = 4×6 − 7×2 = 24 − 14 = 10 (non-zero, so A is invertible). Augmenting with the identity and row-reducing — or using the 2×2 shortcut A⁻¹ = (1/det)×[[d,−b],[−c,a]] — gives A⁻¹ = [[0.6, −0.7], [−0.2, 0.4]]. Checking: A × A⁻¹ = [[4×0.6+7×(−0.2), 4×(−0.7)+7×0.4],[2×0.6+6×(−0.2), 2×(−0.7)+6×0.4]] = [[1, 0], [0, 1]], the identity matrix — confirming the inverse is correct.

Real-World Use Cases

Who Uses Inverse Matrix Calculator and Why

Computing A⁻¹ once so it can be reused to solve Ax = b for several different constant vectors without repeating elimination each time.
Verifying that a manually computed inverse is correct by checking that A × A⁻¹ produces the identity matrix.
Confirming a matrix is invertible (non-singular) before relying on it elsewhere, such as in Cramer's Rule or the inverse matrix method for solving a system.
Learning the Gauss-Jordan augmented-matrix technique by following every row operation shown on [A | I].

Common Mistakes

Mistakes to Avoid

Attempting to invert a matrix with a zero determinant — a singular matrix has no inverse, and no sequence of row operations can make the left half of the augmented matrix become the identity.
Assuming a non-square matrix has a true inverse — only square matrices can have one, and even then only if their determinant is non-zero.
Skipping the A × A⁻¹ = I verification step, which is the most reliable way to catch an arithmetic mistake made anywhere during row reduction.

Pro Tips

Tips for Best Results

Run the Determinant Calculator first if you're unsure whether a matrix is invertible — a non-zero determinant confirms an inverse exists before you invest time computing it.
Always multiply the original matrix by its computed inverse afterward — if the result isn't the identity matrix, an error occurred somewhere in the row operations.

Troubleshooting

Fixing Common Problems

The calculator reports the matrix is singular. — This means the determinant is zero and no inverse exists — at some step in elimination, a pivot column couldn't be made non-zero no matter which row operations were applied.

I need the inverse of a non-square matrix. — True inverses only exist for square matrices; a non-square matrix instead uses a different concept called a pseudoinverse, which this calculator doesn't compute.

Glossary

Terms Explained

Augmented matrix: A matrix formed by placing the identity matrix next to the original matrix A, used to solve for A⁻¹ via row operations.

Pivot: The entry in a column selected to become 1 during row reduction, used to eliminate the rest of that column.

FAQ

Frequently Asked Questions

What does it mean if the calculator says the matrix is singular?

It means the matrix has a determinant of zero and therefore has no inverse — no matter what row operations are applied, at least one pivot column cannot be made non-zero, so the left side can never become the identity matrix.

Why augment the matrix with the identity matrix?

Augmenting with the identity lets you apply the same row operations to both halves at once. Since those operations transform A into I, applying them to the identity side simultaneously transforms it into exactly A⁻¹ — a compact way to solve for the inverse without separate equations.

How can I verify an inverse is correct?

Multiply the original matrix by its computed inverse (in either order) — if the result is the identity matrix (1s on the diagonal, 0s elsewhere), the inverse is correct.

Do only square matrices have inverses?

Yes — only square matrices (same number of rows and columns) can have a true inverse in this sense, and even then only if their determinant is non-zero. Non-square matrices use a different concept called a pseudoinverse, which this calculator does not compute.