The inverse of a square matrix A is the matrix A⁻¹ with A × A⁻¹ = I. For a 2×2 matrix [a, b; c, d], A⁻¹ = 1/(ad − bc) × [d, −b; −c, a]; for larger matrices, row-reduce [A | I] until the left side is I. A matrix with determinant 0 has no inverse. This inverse matrix calculator finds the inverse of 2×2, 3×3 and 4×4 matrices in exact fractions, and also shows the determinant, transpose, adjugate and every Gauss–Jordan step.
How to use the inverse matrix calculator
- Pick the size: 2 × 2, 3 × 3 or 4 × 4.
- Type the entries. Integers, decimals and fractions such as −3/4 all work.
- Read A⁻¹ at the top, with the determinant and an A × A⁻¹ = I check.
- Scroll for the transpose and adjugate, and open the steps to follow the row operations.
- Switch to decimals if you need them, and copy the link to share the matrix.
2×2 inverse formula
For A = [a, b; c, d], OpenStax gives A⁻¹ = 1/(ad − bc) × [d, −b; −c, a]. Example: A = [4, 7; 2, 6].
- Determinant: 4 × 6 − 7 × 2 = 10.
- Swap the diagonal and negate the off-diagonal: [6, −7; −2, 4].
- Divide by 10: A⁻¹ = [3/5, −7/10] [−1/5, 2/5].
Check: A × A⁻¹ = [1, 0] [0, 1], the identity. Another quick one: [3, 1; 5, 2] has determinant 1, so its inverse is just [2, −1] [−5, 3] — whole numbers whenever the determinant is ±1.
3×3 inverse by Gauss–Jordan elimination
Take A = [1, 2, 3] [0, 1, 4] [5, 6, 0]. The calculator's row operations on [A | I]:
- Write the augmented matrix [A | I].
- Clear column 1: R3 → R3 − 5·R1.
- Clear column 2: R1 → R1 − 2·R2; R3 → R3 + 4·R2.
- Clear column 3: R1 → R1 + 5·R3; R2 → R2 − 4·R3.
- The left side is now I, so the right side is A⁻¹.
Result:
| −24 | 18 | 5 |
| 20 | −15 | −4 |
| −5 | 4 | 1 |
The determinant is 1, which is why every entry is a whole number. Multiplying back gives the identity matrix exactly.
3×3 inverse by the adjugate method
The same inverse comes from cofactors. Each cofactor Cij = (−1)i+j × the 2×2 determinant left after deleting row i and column j. For the matrix above the cofactor matrix is [−24, 20, −5] [18, −15, 4] [5, −4, 1]; transposing it gives the adjugate [−24, 18, 5] [20, −15, −4] [−5, 4, 1], and dividing by det(A) = 1 gives A⁻¹. This method is neat by hand for 3×3 matrices; for 4×4 and larger, elimination is far less work.
A 4×4 example
A = [2, 1, 1, 0] [4, 3, 3, 1] [8, 7, 9, 5] [6, 7, 9, 8] has determinant 8 and inverse
| 9/4 | −3/4 | −1/4 | 1/4 |
| −3 | 5/2 | −1/2 | 0 |
| −1/2 | −1 | 1 | −1/2 |
| 3/2 | −1/2 | −1/2 | 1/2 |
Fractions appear because the determinant is 8: every entry of the adjugate is divided by it.
Transpose, determinant and inverse rules
| Rule | Meaning |
|---|---|
| (Aᵀ)ᵀ = A | Transposing twice returns the original |
| (A⁻¹)⁻¹ = A | The inverse of the inverse is the original |
| (AB)⁻¹ = B⁻¹A⁻¹ | Inverting a product reverses the order |
| (Aᵀ)⁻¹ = (A⁻¹)ᵀ | Transpose and inverse can be done in either order |
| det(Aᵀ) = det(A) | The transpose has the same determinant |
| det(A⁻¹) = 1 / det(A) | So a zero determinant means no inverse |
These are standard results covered in MIT's 18.06 Linear Algebra lectures. For the 2×2 example, det(A⁻¹) = 1/10 = 1 ÷ 10.
Singular matrices
If a row is a multiple of another, or more generally the rows are linearly dependent, elimination produces a row of zeros on the left and no pivot can be found. [1, 2; 2, 4] is the simplest example: det = 1 × 4 − 2 × 2 = 0. The calculator reports the column where elimination fails. A determinant that is merely very small in decimal input (such as 0.0000001) is a different case: the matrix is invertible but nearly singular, and its inverse has very large entries.
Common mistakes
- Swapping the wrong entries in the 2×2 formula. Swap the main diagonal (a and d) and negate b and c — not the other way round.
- Forgetting the checkerboard signs. Cofactors alternate + − + across each row, starting with + in the top-left.
- Not transposing the cofactor matrix. The adjugate is the transpose of the cofactors.
- Row operations on only one side. Every operation must be applied to the whole row of [A | I].
- Rounding mid-way. Keep fractions exact until the end.
More maths tools: vertex form calculator, fraction calculator, interpolation calculator and average and standard deviation.
Frequently asked questions
How do you find the inverse of a 2×2 matrix?
Swap the two diagonal entries, change the signs of the other two, and divide every entry by the determinant ad − bc. For [4, 7; 2, 6] the determinant is 4 × 6 − 7 × 2 = 10, so the inverse is [3/5, −7/10] [−1/5, 2/5].
How do you find the inverse of a 3×3 matrix?
Write the augmented matrix [A | I] and use row operations — swap rows, multiply a row by a non-zero number, add a multiple of one row to another — until the left side becomes the identity. The right side is then A⁻¹. This is the Gauss–Jordan method in OpenStax College Algebra; the calculator lists every operation.
When does a matrix have no inverse?
When its determinant is 0. Such a matrix is called singular: its rows (or columns) are linearly dependent, so elimination produces a row of zeros. [1, 2; 2, 4] is singular because the second row is twice the first; its determinant is 0.
What is the determinant?
A single number computed from a square matrix. For a 2×2 matrix it is ad − bc; larger determinants expand along a row using cofactors. A non-zero determinant means the matrix is invertible, and det(A⁻¹) = 1 ÷ det(A).
What is the transpose of a matrix?
The matrix you get by turning rows into columns: entry (i, j) moves to (j, i). A 2 × 3 matrix [1, 2, 3] [4, 5, 6] has transpose [1, 4] [2, 5] [3, 6]. Any matrix has a transpose, square or not, and the inverse of the transpose is the transpose of the inverse.
What is the adjugate (adjoint) matrix?
The transpose of the cofactor matrix. Each cofactor is (−1)^(i+j) times the determinant of the matrix left after deleting row i and column j. Dividing the adjugate by the determinant gives the inverse: A⁻¹ = adj(A) ÷ det(A).
How do you check that an inverse is correct?
Multiply: A × A⁻¹ should equal the identity matrix I (ones on the diagonal, zeros elsewhere), and so should A⁻¹ × A. The calculator does this check with exact fractions.
Why use fractions instead of decimals?
Inverses usually contain fractions such as 3/5 or −7/10. Rounding them to decimals can make A × A⁻¹ come out slightly off the identity. Exact arithmetic avoids that; switch to decimals only for display.
How are inverse matrices used to solve equations?
A system of linear equations can be written AX = B. If A is invertible, X = A⁻¹B, which is how OpenStax solves systems with inverses. Multiply the inverse from this calculator by your constants column to get the solution.
Do non-square matrices have inverses?
No. Only square matrices (2×2, 3×3 and so on) can have a two-sided inverse, and even then only when the determinant is not zero.
Sources & method
- OpenStax College Algebra 2e §7.7 — Solving systems with inverses (identity matrix, 2×2 formula, Gauss–Jordan on [A | I])
- OpenStax College Algebra 2e §7.8 — Solving systems with Cramer's rule (2×2 and 3×3 determinants)
- OpenStax College Algebra 2e §7.5 — Matrices and matrix operations (matrix multiplication)
- MIT OpenCourseWare 18.06 Linear Algebra (Strang) — Lecture 3: multiplication and inverse matrices; Lecture 20: Cramer's rule, inverse matrix
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