Math
Matrix Inverse Calculator
Enter a square matrix to get its inverse exactly, as reduced fractions with a decimal view.
The panel walks through Gauss-Jordan elimination on the augmented matrix [A | I], lists every row operation, shows det(A) and (for 2 x 2 and 3 x 3) the adjugate formula A⁻¹ = adj(A) / det(A), then verifies A·A⁻¹ = I. If the matrix is singular you get a clear explanation instead of a wrong answer.
One row per line, entries separated by spaces or commas. Must be square. Fractions like 1/2 stay exact in the inverse.
Try an example
Result
Inverse matrix
—
- det(A)
- —
- Rank
- —
- Verification
- —
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What the inverse of a matrix is
The inverse of a square A rectangular grid of numbers arranged in rows and columns. A is the matrix A⁻¹ with A·A⁻¹ = A⁻¹·A = I, the identity matrix. Multiplying by A⁻¹ undoes multiplying by A, which is how you solve Ax = b: x = A⁻¹b. An inverse exists exactly when det(A) ≠ 0, or equivalently when the rows are linearly independent (full rank).
Inverse formulas
How to use it
- Type the matrix, one row per line.
- Read A⁻¹ in the panel. Fractions are exact; a decimal version appears when entries are not integers.
- Open the Gauss-Jordan steps to follow each row operation on [A | I], and Show the work for the One number computed from a square matrix that tells you whether it can be inverted. and the check.
- If the panel says Describes a square matrix whose determinant is zero, so it has no inverse., det(A) = 0: the matrix has no inverse, and no amount of row reduction will produce one.
How to read the answer
- The right half of the reduced augmented matrix is A⁻¹. The left half must be exactly the identity.
- det(A⁻¹) = 1 / det(A), so a matrix with a small determinant has an inverse with large entries.
- The check A·A⁻¹ = I is computed exactly, so a verified result is correct to every digit.
When a matrix has no inverse
| Symptom | What it means |
|---|---|
| det(A) = 0 | Singular; the inverse does not exist |
| A row of zeros appears on the left of [A | I] | Rank is less than n; elimination cannot finish |
| Two rows or columns are proportional | Linear dependence; determinant is 0 |
| Matrix is not square | Inverse is not defined at all (see rank or pseudo-inverse instead) |
Common mistakes
- Swapping a and d but forgetting to negate b and c in the 2 x 2 formula.
- Dividing by ad - bc when it is 0. The formula only applies to Describes a matrix that has an inverse, which happens when its determinant is not zero. matrices.
- Assuming (AB)⁻¹ = A⁻¹B⁻¹. The correct rule is (AB)⁻¹ = B⁻¹A⁻¹, order reversed.
- Rounding the inverse to two decimals and then checking A·A⁻¹; the small errors compound. Use the exact fractions.
How to type a matrix into this calculator
- Put each row on its own line. Separate the entries in a row with spaces or commas: 2 1 1 or 2, 1, 1.
- Fractions such as 1/3 and decimals such as 0.25 are accepted and kept exact, so 1/3 stays 1/3 instead of becoming 0.3333.
- Every row must have the same number of entries. A row with a missing entry produces a clear error instead of a wrong answer.
- The largest size supported is 8 x 8. Brackets and semicolons are ignored, so [1 2; 3 4] works too.
Worked examples
3 x 3 inverse
A = [2 1 1; 1 3 2; 1 0 0], det = -1
Inverse matrix
[0, 0, 1; -2, 1, 3; 3, -1, -5]
2 x 2 with fractions in the answer
A = [4 7; 2 6] → A⁻¹ = [3/5 -7/10; -1/5 2/5]
Inverse matrix
[3/5, -7/10; -1/5, 2/5]
Singular matrix
Rows are proportional, so no inverse exists
Inverse matrix
No inverse (singular)
Identity inverts to itself
I⁻¹ = I
Inverse matrix
[1, 0, 0; 0, 1, 0; 0, 0, 1]
Decimal entries
A = [0.5 0.25; 1 2] is inverted exactly
Inverse matrix
[8/3, -1/3; -4/3, 2/3]
Not square
A 2 x 3 matrix cannot be inverted
Inverse matrix
Error
Frequently asked questions
How do I know if a matrix has an inverse?+
Compute the determinant. If det(A) ≠ 0 the inverse exists; if det(A) = 0 the matrix is singular and has no inverse. The calculator reports both.
What is the fastest way to invert a 2 x 2 matrix by hand?+
Swap the diagonal entries, negate the off-diagonal entries, and divide everything by ad - bc. For [4 7; 2 6] that gives (1/10)·[6 -7; -2 4].
Why does Gauss-Jordan on [A | I] give the inverse?+
Each row operation is multiplication by an elementary matrix. Reducing A to I means multiplying by E_k…E_1 = A⁻¹, and the same product applied to I on the right side leaves A⁻¹ there.
Can a non-square matrix have an inverse?+
Not a true two-sided inverse. Rectangular matrices can have left or right inverses or a Moore-Penrose pseudo-inverse, but A⁻¹ in the usual sense requires a square matrix.
Is the inverse of a matrix with integer entries always integer?+
No. Entries of A⁻¹ are cofactors divided by det(A), so they are integers only when det(A) = ±1. The calculator keeps them as exact fractions.
How do I use the inverse to solve a system?+
For Ax = b, multiply both sides by A⁻¹: x = A⁻¹b. For a single system, row reduction (RREF) is usually fewer steps; the inverse pays off when you solve many systems with the same A.
About this calculator
- Written by
- mathcheck editorial team
- Last reviewed
- September 4, 2026
Method
- Uses the values entered by the user and stable formulas documented on the page.
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Last updated: September 4, 2026