Math
Adjugate Matrix Calculator
Enter a square matrix to get its cofactor matrix and adjugate, also called the classical adjoint.
The calculator computes each cofactor as a signed minor, transposes to form adj(A), verifies A·adj(A) = det(A)·I exactly, and, when det(A) ≠ 0, divides to give the inverse.
One row per line. Separate entries with spaces or commas. Integers, decimals, and fractions like 1/3 all work. Up to 8 x 8.
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Result
Adjugate
—
- Cofactor matrix
- —
- det(A)
- —
- Inverse
- —
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Minors, cofactors, and the adjugate
The minor M_ij is the One number computed from a square matrix that tells you whether it can be inverted. of the A rectangular grid of numbers arranged in rows and columns. left after deleting row i and column j. The A smaller determinant with a plus or minus sign attached, used to expand a bigger matrix. C_ij attaches the checkerboard sign (-1)^(i+j) to that minor. The matrix of all cofactors is the cofactor matrix; its The matrix you get by flipping rows into columns and columns into rows. is the adjugate adj(A), also called the classical adjoint.
The cofactor sign pattern
| Row \ Column | 1 | 2 | 3 | 4 |
|---|---|---|---|---|
| 1 | + | - | + | - |
| 2 | - | + | - | + |
| 3 | + | - | + | - |
| 4 | - | + | - | + |
How to use it
- Enter a square matrix, one row per line.
- Read the cofactor matrix and adj(A) in the panel; open Show the work to see each signed minor.
- Use the verification A·adj(A) = det(A)·I to confirm, and take A⁻¹ = adj(A)/det(A) when the determinant is nonzero.
How to read the answer
- For a 2 x 2 matrix [a b; c d], adj(A) = [d -b; -c a]: swap the diagonal, negate the off-diagonal.
- The adjugate exists for every square matrix, even Describes a square matrix whose determinant is zero, so it has no inverse. ones; the inverse does not.
- det(adj(A)) = det(A)^(n-1), so the adjugate of a singular matrix (n ≥ 2) is itself singular.
Common mistakes
- Forgetting to transpose. The adjugate is the transpose of the cofactor matrix, not the cofactor matrix itself.
- Applying the sign pattern to the original entries instead of the minors.
- Using the 2 x 2 swap-and-negate shortcut on a 3 x 3 matrix; there every entry needs a 2 x 2 minor.
- Confusing adjugate with the conjugate transpose (also called adjoint in some contexts); they are different objects.
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 adjugate
A = [1 2 3; 0 1 4; 5 6 0], det = 1, so adj(A) = A⁻¹
Adjugate
[-24, 18, 5; 20, -15, -4; -5, 4, 1]
2 x 2 shortcut
adj [a b; c d] = [d -b; -c a]
Adjugate
[6, -7; -2, 4]
Singular matrix still has an adjugate
A = [1 2; 2 4] → adj = [4 -2; -2 1], A·adj = 0
Adjugate
[4, -2; -2, 1]
Identity
adj(I) = I
Adjugate
[1, 0, 0; 0, 1, 0; 0, 0, 1]
Fractions
A = [1/2 1; 1/3 2]
Adjugate
[2, -1; -1/3, 1/2]
Not square
No cofactors for a 2 x 3 matrix
Adjugate
Error
Frequently asked questions
What is the difference between the adjugate and the adjoint?+
In most linear algebra courses 'adjoint' means the classical adjoint, which is the adjugate (transpose of the cofactor matrix). In functional analysis 'adjoint' means the conjugate transpose. This page computes the adjugate.
How do I find the adjugate of a 3 x 3 matrix?+
Compute all nine 2 x 2 minors, apply the checkerboard signs to get the cofactors, arrange them in a matrix, and transpose it.
Why does A·adj(A) equal det(A)·I?+
Each diagonal entry of the product is a cofactor expansion of det(A). Each off-diagonal entry is the expansion of a matrix with two equal rows, which is 0.
Can I use the adjugate to find the inverse?+
Yes: A⁻¹ = adj(A)/det(A). It is a clean formula for 2 x 2 and 3 x 3 matrices; for larger matrices Gauss-Jordan elimination is less work.
Does a singular matrix have an adjugate?+
Yes. The adjugate only needs minors, so it exists for every square matrix. For a singular matrix A·adj(A) is the zero matrix.
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