Enter a square matrix up to 8 × 8 to get its inverse in exact fractions. Every row operation of Gauss–Jordan elimination (or the adjugate method) is shown, singular matrices are detected and explained, and the result is verified by multiplying back to the identity.
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Matrix A
Entries can be integers, decimals or fractions like -3/4 or 1 1/2. Empty cells count as 0. Arrow keys move between cells.
A square matrix has an inverse exactly when its determinant is not 0.
Inverse
Step by step
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How to use the Matrix Inverse Calculator
Choose the size of the matrix (2 × 2 to 8 × 8).
Enter the entries – integers, decimals or fractions – or paste rows with Paste rows. Random and Identity fill the grid quickly.
Pick the method: Gauss–Jordan elimination or the adjugate formula.
Press Calculate inverse. The inverse also updates as you type.
Follow the steps and copy the result as LaTeX, CSV or for a spreadsheet.
Worked example: inverting a 3 × 3 matrix
Take A = [[2, 1, 1], [1, 3, 2], [1, 0, 0]] (the default matrix above). Its determinant is −1, so it is invertible. Gauss–Jordan elimination on [A | I] gives
Column 1
Column 2
Column 3
Row 1 of A⁻¹
0
0
1
Row 2 of A⁻¹
−2
1
3
Row 3 of A⁻¹
3
−1
−5
Check one entry: row 1 of A times column 1 of A⁻¹ is 2·0 + 1·(−2) + 1·3 = 1, the (1,1) entry of the identity.
Properties of the inverse
Property
Rule
Definition
A · A⁻¹ = A⁻¹ · A = I
Inverse of a product
(AB)⁻¹ = B⁻¹A⁻¹ (order reverses)
Inverse of a transpose
(Aᵀ)⁻¹ = (A⁻¹)ᵀ
Inverse of a scalar multiple
(kA)⁻¹ = (1/k)·A⁻¹, k ≠ 0
Determinant
det(A⁻¹) = 1 / det(A)
Diagonal matrix
Invert each diagonal entry
Orthogonal matrix
A⁻¹ = Aᵀ
Double inverse
(A⁻¹)⁻¹ = A
Gauss–Jordan or adjugate?
Gauss–Jordan elimination needs roughly n³ arithmetic operations, so it stays practical for 5 × 5 to 8 × 8 matrices. The adjugate formula needs n² determinants of size n − 1, which grows much faster; it is mainly useful for 2 × 2 and 3 × 3 matrices and for proving properties such as A⁻¹ = adj(A)/det(A). Both give the same exact answer here.
Frequently asked questions
How do you find the inverse of a matrix?
Place the identity matrix next to A to form [A | I] and apply row operations (swap rows, multiply a row by a non-zero number, add a multiple of one row to another) until the left half becomes I. The right half is then A⁻¹. This is Gauss–Jordan elimination and works for any size.
What is the formula for the inverse of a 2×2 matrix?
For A = [[a, b], [c, d]], A⁻¹ = 1/(ad − bc) · [[d, −b], [−c, a]]. Swap the diagonal entries, change the sign of the off-diagonal entries and divide by the determinant ad − bc. If ad − bc = 0 there is no inverse.
When does a matrix have no inverse?
A square matrix is not invertible (it is singular) when its determinant is 0. Equivalently its rank is less than its size, its rows or columns are linearly dependent, and Gauss–Jordan elimination runs into a column with no usable pivot. Non-square matrices never have a two-sided inverse.
What is the adjugate method?
A⁻¹ = adj(A) / det(A), where adj(A) is the transpose of the cofactor matrix and each cofactor is Cᵢⱼ = (−1)^(i+j) times the determinant of A with row i and column j removed. It is convenient for 2 × 2 and 3 × 3 matrices; for larger ones Gauss–Jordan elimination is much faster by hand.
Why are the answers fractions?
The inverse of a matrix with integer entries usually has fractions with the determinant as a common denominator. The calculator uses exact big-integer fraction arithmetic, so there is no rounding error even for 8 × 8 matrices. Choose Decimals to see decimal approximations.
How can I check an inverse?
Multiply the matrix by its inverse: A · A⁻¹ and A⁻¹ · A must both equal the identity matrix. The calculator performs this multiplication exactly and shows a check mark when it holds.
What is the inverse used for?
Solving linear systems Ax = b as x = A⁻¹b, undoing linear transformations in graphics and robotics, computing least-squares regression ((XᵀX)⁻¹Xᵀy) and changing coordinate bases. For a single system, elimination on [A | b] is usually faster than computing the full inverse.
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