Inverse Matrix Calculator
Enter your square matrix values and this calculator finds the inverse matrix and the determinant. Choose a 2x2 or 3x3 matrix, fill in the entries, and get the inverse with a full worked solution showing each cofactor, the adjugate, and the final result. The calculator also tells you if the matrix is singular (non-invertible) and why.
What is the inverse of a matrix?
The inverse of a square matrix A is another matrix, written A⁻¹, such that A multiplied by A⁻¹ equals the identity matrix I: a matrix with 1s on the main diagonal and 0s everywhere else. Not every matrix has an inverse: a matrix only has one when its determinant is non-zero. Such matrices are called invertible, non-singular, or regular. If the determinant is zero the matrix is singular and cannot be inverted. Matrix inverses are used widely: solving systems of linear equations, computing transformations in computer graphics and robotics, inverting covariance matrices in statistics, and solving differential equations in engineering.
How to find the inverse - the adjugate method
The most direct formula uses the adjugate (also called the classical adjoint). For any invertible matrix A: A⁻¹ = (1 / det(A)) * adj(A) The adjugate is the transpose of the cofactor matrix. Each cofactor C_ij equals (-1)^(i+j) times the determinant of the (n-1)x(n-1) submatrix obtained by deleting row i and column j. For a 2x2 matrix A = [[a, b], [c, d]]: det(A) = ad - bc, and A⁻¹ = (1/(ad-bc)) * [[d, -b], [-c, a]]. The entries of the adjugate are formed by swapping the main-diagonal elements and changing the sign of the off-diagonal elements. For a 3x3 matrix you compute 9 cofactors (each requiring a 2x2 determinant), arrange them into the cofactor matrix, transpose to get the adjugate, and divide by the determinant.
Gauss-Jordan elimination as an alternative
An alternative method that scales better for larger matrices is Gauss-Jordan elimination. You augment the n x n matrix A with an n x n identity matrix to form [A | I], then apply row operations to reduce the left side to the identity. When the left half becomes I, the right half becomes A⁻¹. Row operations allowed are: swapping two rows, multiplying a row by a non-zero scalar, and adding a multiple of one row to another. If at any point you cannot reduce the left side fully (a row becomes all zeros), the matrix is singular and has no inverse. This method is equivalent mathematically to the adjugate formula but is often easier to implement for larger matrices.
Key properties of matrix inverses
Several identities are useful when working with inverses: - (A⁻¹)⁻¹ = A: inverting twice returns the original matrix. - (AB)⁻¹ = B⁻¹ A⁻¹: the inverse of a product reverses the order of the factors. - (Aᵀ)⁻¹ = (A⁻¹)ᵀ: the inverse of the transpose equals the transpose of the inverse. - det(A⁻¹) = 1 / det(A): the determinant of the inverse is the reciprocal of the determinant. - A matrix and its inverse share the same eigenvectors; the eigenvalues of A⁻¹ are the reciprocals of the eigenvalues of A. These properties are fundamental when manipulating matrix equations and proofs in linear algebra.
Matrix types and invertibility
| Matrix type | Determinant | Invertible? | Notes |
|---|---|---|---|
| Identity (I) | 1 | Yes | Its own inverse: I⁻¹ = I |
| Diagonal (all non-zero diag) | Product of diag entries | Yes | Inverse: reciprocals on diagonal |
| Singular | 0 | No | Rows/columns are linearly dependent |
| Orthogonal (Q) | ±1 | Yes | Q⁻¹ = Qᵀ (transpose equals inverse) |
| Upper/Lower triangular | Product of diag | Yes (if diag non-zero) | Inverse also triangular |
| Scalar multiple of I (kI) | kⁿ | Yes (if k ≠ 0) | (kI)⁻¹ = (1/k)I |
| Zero matrix | 0 | No | Every row is zero |
Quick reference for common matrix forms and whether their inverse exists.
Frequently asked questions
What does it mean if the determinant is zero?
A zero determinant means the matrix is singular: its rows (or equivalently its columns) are linearly dependent, so the matrix squishes space into a lower dimension and cannot be reversed. For example, if one row is a multiple of another, the matrix maps two different input vectors to the same output, making it impossible to reverse. This also means the system of equations Ax = b either has no solution or infinitely many solutions.
How do I verify the inverse is correct?
Multiply the original matrix A by its computed inverse A⁻¹. The result should be the identity matrix I (1s on the main diagonal, 0s everywhere else). If you get I, the calculation is correct. Small rounding errors are normal in floating-point arithmetic; entries close to zero (like 1e-15) are effectively zero.
What is the difference between the adjugate method and Gauss-Jordan elimination?
Both methods give the same result. The adjugate (cofactor) method uses an explicit formula, A⁻¹ = adj(A) / det(A), and is easiest to apply by hand for 2x2 and 3x3 matrices. Gauss-Jordan elimination augments A with the identity and applies row operations; it is more systematic for 4x4 and larger matrices and is what most computer libraries use under the hood.
Can a non-square matrix have an inverse?
A true (two-sided) inverse only exists for square matrices. Non-square matrices can have a pseudoinverse (the Moore-Penrose inverse), which gives the least-squares solution to over- or under-determined systems. This calculator handles square 2x2 and 3x3 matrices.
Why does inverting twice give the original matrix?
If B = A⁻¹, then by definition A × B = I, which also means B × A = I, so B is invertible and A is its inverse: B⁻¹ = A. In other words (A⁻¹)⁻¹ = A. This is analogous to how dividing twice by the same non-zero number returns the original value.
Is the inverse of a matrix unique?
Yes. If A has an inverse, that inverse is unique. Suppose B and C are both inverses of A. Then B = B(AC) = (BA)C = IC = C, so B and C must be the same matrix.