Inverse Matrix Calculator

1. Choose the matrix size, 2 by 2 or 3 by 3.
2. Enter the numbers row by row.
3. Read the inverse matrix, shown row by row.
4. Check the determinant and whether an inverse exists.

Here is what each result means:

The inverse of a matrix is another matrix that undoes it. When you multiply a matrix by its inverse, you get the identity matrix, which acts like the number 1 in matrix multiplication. It is written as A to the power minus one, and it plays the role of dividing by a matrix.

Only a square matrix with a nonzero determinant has an inverse. If the determinant is zero, the matrix is singular and no inverse exists, just as you cannot divide by zero. Inverses are used to solve systems of equations, reverse transformations and much more.

An inverse, when it exists, is unique, and it works from either side: the original times the inverse and the inverse times the original both give the identity. Geometrically, if the matrix stretches, rotates or shears space, the inverse applies exactly the opposite motion, returning every point to where it started.

It divides the adjugate, the transposed matrix of cofactors, by the determinant.

1. Find the determinant; if it is zero, there is no inverse.
2. Build the matrix of cofactors and transpose it for the adjugate.
3. Divide every entry of the adjugate by the determinant.

The determinant is the gatekeeper here; you can check it first with the determinant calculator.

Find the inverse of the 2x2 matrix with rows 4, 7 and 2, 6.

Calculation: the determinant is 4 x 6 - 7 x 2 = 24 - 14 = 10. Swap the diagonal, negate the other two, and divide by 10: the inverse is [0.6, -0.7 ; -0.2, 0.4]. Multiplying the original by this gives the identity.

A 2x2 inverse has a quick recipe once you have the determinant.

The 3x3 case follows the same idea but uses cofactors, which is why it takes more steps.

Whether an inverse exists comes down to the determinant.

A singular matrix squashes space onto a line or plane, so the transformation cannot be undone.

The Determinant

A zero determinant means no inverse. A small determinant can make the inverse entries large.

Dependent Rows

If rows are multiples or sums of each other, the determinant is zero and no inverse exists.

The Size

A 3x3 inverse needs cofactors and far more arithmetic than a 2x2.

Solving Linear Systems

Multiply by the inverse to solve a system written in matrix form.

Reversing Transformations

Undo a rotation, scaling or other linear map.

Checking Invertibility

See at a glance whether a matrix can be inverted.

1. Inverting a Singular Matrix

A zero determinant means no inverse exists. Check the determinant first.

2. Forgetting to Divide by the Determinant

The adjugate alone is not the inverse. Every entry must be divided by the determinant.

3. Skipping the Transpose

The adjugate is the transpose of the cofactor matrix. Missing the transpose gives a wrong answer.

4. Sign Errors in Cofactors

Cofactors alternate in sign. A single sign slip changes the inverse.

5. Expecting a Times a to Be the Identity

Only A times its inverse gives the identity, not A times itself.

The method is exact; only the displayed decimals are rounded.

It is a matrix that undoes another: multiplying a matrix by its inverse gives the identity matrix, which acts like 1 in matrix multiplication. It is written A to the power minus one.

Find the determinant, a d minus b c. Then swap the two diagonal entries, negate the other two, and divide every entry by the determinant. This tool does that automatically.

When its determinant is zero. Such a matrix is called singular. It squashes space onto a lower dimension, so the transformation cannot be reversed.

Compute the determinant, build the matrix of cofactors, transpose it to get the adjugate, and divide every entry by the determinant. The steps are longer but follow the same idea as a 2x2.

It is a square matrix with ones on the diagonal and zeros elsewhere. Multiplying any matrix by its inverse gives this identity, which leaves other matrices unchanged.

The inverse formula divides by the determinant, so a determinant of zero makes it undefined. Only a nonzero determinant allows an inverse to exist.

It is the transpose of the cofactor matrix. Dividing the adjugate by the determinant gives the inverse. It carries the sign pattern of the cofactors.

They solve systems of linear equations, reverse transformations like rotations and scalings, and appear throughout linear algebra, graphics and statistics.

No. The calculation runs in your browser and nothing you enter is stored or sent anywhere, unless you choose Save, which keeps the result only on this device.

• Invertible matrix (Wikipedia).
• Inverse of a matrix (Maths Is Fun).
• Adjugate matrix (Wikipedia).