Inverse Matrix Calculator

Find A⁻¹ with exact row operations, detect a singular matrix, and verify the result with the identity.

Drag & drop or upload an image or PDF

Enter a square matrix, such as inverse([[2,1],[5,3]])

Enter rows in one line, such as [[1, 2], [3, 4]]. State the operation when the page accepts more than one.

Try an example

An inverse undoes a square matrix transformation

A matrix inverse satisfies AA⁻¹ = A⁻¹A = I. It exists only for a square, nonsingular matrix. A determinant of zero or a missing pivot during row reduction means there is no ordinary inverse.

Gauss-Jordan elimination works by applying every row operation to the augmented matrix [A | I]. When the left side becomes the identity, the right side is A⁻¹. For a 2 × 2 matrix, the determinant formula is often quicker.

How to use the inverse matrix calculator

Enter the problem as written

Type or paste the full expression. You can also upload a clear photo or PDF and check the extracted text before solving.

Read the working, not only the answer

Each transformation is separated and explained so you can compare it with your own method.

Ask about any step

Continue in the same solution to request another method, check a restriction, or ask why a rule applies.

Inverse of a 2 × 2 matrix

Swap the diagonal entries, negate the off-diagonal entries, and divide by the nonzero determinant.

[a c b d]^{- 1} = \frac{1}{a d - b c} [d - c - b a]

A reliable way to work through it

Check invertibility

Confirm the matrix is square and that its determinant is nonzero before applying an inverse formula.

Apply every row operation to both sides

In [A | I], each swap, scaling, or row replacement must affect the entire augmented row.

Multiply to verify

A correct result produces the identity when multiplied by A in either order.

Worked example

Find the inverse of A

A = [2513], det (A) = 2 (3) - 1 (5) = 1

The determinant is nonzero, so the inverse exists.

A^{- 1} = \frac{1}{1} [3 - 5 - 1 2]

Apply the 2 × 2 inverse formula.

A A^{- 1} = [1001]

Multiply to confirm the identity matrix.

The inverse has rows [3, -1] and [-5, 2].

Common mistakes to check

Taking entrywise reciprocals

A matrix inverse is not found by replacing each entry with its reciprocal.

Dividing by a zero determinant

A singular matrix has no ordinary inverse, so the method must stop rather than divide by zero.

Changing only half the augmented matrix

Every row operation applied to A must also be applied to the identity side.

Determinant CalculatorCalculate the scalar determinant of a square matrix and track every cofactor sign or row-operation effect.Matrix CalculatorPerform matrix operations with dimension checks, exact entries, and each row or column calculation shown.Systems of Equations CalculatorSolve two or more equations together with substitution, elimination, or row operations.Eigenvalue CalculatorFind the characteristic polynomial, eigenvalues, and eigenspaces of a square matrix.

Questions students ask

Does every square matrix have an inverse?

No. An ordinary inverse exists exactly when the determinant is nonzero.

How can I check an inverse?

Multiply A and A⁻¹ in either order and confirm that the result is the identity matrix.

Why can an inverse contain fractions?

Row operations and division by the determinant naturally produce rational entries.

Is (AB)⁻¹ equal to A⁻¹B⁻¹?

No. The order reverses: (AB)⁻¹ = B⁻¹A⁻¹.