Inverse Matrix Calculator

What this calculator does

The inverse of a matrix is its undo button. Multiply a matrix by its inverse and you get back to the starting point, the matrix equivalent of the number 1. This finds it for 2 by 2, 3 by 3 and 4 by 4 matrices.

Pick the size, enter the numbers, and it returns the inverse. It runs right here in the browser.

Using the calculator

1. Choose the matrix size: 2x2, 3x3 or 4x4.
2. Enter the numbers row by row.
3. Press Calculate.

It returns the inverse matrix, or tells you if none exists. Reset clears the boxes.

What an inverse is

With ordinary numbers, the inverse of 5 is one fifth, because 5 times one fifth is 1. Matrices have the same idea. The inverse of a matrix A, written A^-1, is the matrix that, multiplied by A, gives the identity matrix, which is the matrix version of the number 1, with 1s down its diagonal and 0s everywhere else. In symbols, A × A^-1 = I.

As with the determinant, only square matrices can have an inverse. And even among square matrices, not all of them do, which is the next thing to understand.

The 2 by 2 formula

For a 2 by 2 matrix [[a, b], [c, d]], the inverse has a memorable recipe:

A^-1 = (1 ÷ det) × [[d, -b], [-c, a]]

In words: swap the two numbers on the main diagonal, put a minus sign in front of the other two, and then divide every entry by the determinant, ad minus bc. That division by the determinant is the crux of the whole thing, and it is where inverses can fall apart.

When there is no inverse

Look again at that formula and you can see the catch: it divides by the determinant. If the determinant is zero, you would be dividing by zero, which is impossible, so the matrix simply has no inverse. Matrices like this are called singular.

This is not a quirk of the formula, it reflects something real. A determinant of zero means the matrix flattens space onto a line or a point, throwing information away, and once that is gone there is no way to reconstruct what you started with. There is no undo button because the original cannot be recovered. So the rule is simple and worth remembering: a matrix has an inverse exactly when its determinant is not zero.

A worked example

Invert [[4, 7], [2, 6]].

1. Determinant: (4 × 6) − (7 × 2) = 24 − 14 = 10. Not zero, so an inverse exists.
2. Swap the diagonal and negate the rest: [[6, -7], [-2, 4]].
3. Divide every entry by 10: [[0.6, -0.7], [-0.2, 0.4]].

You can check it: multiply the original by this result and you get [[1, 0], [0, 1]], the identity, confirming it really is the inverse.

Why inverses matter

Inverses are how you divide by a matrix, and that unlocks two big jobs. First, solving systems of equations: a whole system can be written as A times x equals b, and multiplying by the inverse gives x equals A^-1 times b, the solution in one move. Second, reversing a transformation: if a matrix rotates or scales space, its inverse rotates or scales it straight back. Anywhere a matrix does something you need to undo, its inverse is the tool.

Bigger matrices

The neat swap-and-divide trick is special to the 2 by 2 case. For 3 by 3 and 4 by 4 matrices, the inverse is found with more work, either by building it from the matrix of cofactors and the determinant, or by a step-by-step elimination process. The guiding rule is unchanged, though: an inverse exists only when the determinant is not zero, and that is what the tool checks before handing you an answer.

Questions people ask

What is the inverse of a matrix?

The matrix A^-1 that, multiplied by A, gives the identity matrix, the matrix equivalent of 1. It acts as an undo for the original.

How do you invert a 2 by 2 matrix?

Swap the main-diagonal entries, negate the other two, and divide every entry by the determinant, ad minus bc.

When does a matrix have no inverse?

When its determinant is zero. Such matrices are called singular, because dividing by that zero determinant is impossible and the transformation cannot be reversed.

What are inverses used for?

Solving systems of equations, written as x equals A^-1 times b, and reversing transformations such as rotations and scalings.

Can a non-square matrix have an inverse?

Not in the usual sense. Only square matrices, and only those with a non-zero determinant, have a true inverse.

References

A note on where this comes from. The inverse of a square matrix A is the matrix A^-1 with A times A^-1 equal to the identity, and it exists precisely when the determinant is non-zero. The first appearance of a matrix inverse is credited to Arthur Cayley in 1853. For further reading, see Invertible matrix.

1. The matrix inverse, A^-1, defined by A times A^-1 equal to the identity, existing only when the determinant is non-zero.
2. Arthur Cayley, credited with the first statement of a matrix inverse (1853).