Cofactor Matrix & Adjugate (Adjoint) Calculator

Background.

The cofactor matrix and the adjugate — the object school textbooks call the classical adjoint, adj(A) — come out of the same short computation, and this calculator does all of it: the matrix of minors, the cofactor matrix once the checkerboard signs are applied, and the adjugate as the transpose of that. It handles any square matrix from 1 by 1 up to 6 by 6.

Type or paste your matrix one row per line, entries separated by commas or spaces. Semicolons work as row separators and the bracketed form [[1,2],[3,4]] is accepted too.

The chain is short and each step has a name. The minor M(i,j) is the determinant of what is left after you delete row i and column j. The cofactor C(i,j) is the same number with the sign (-1) to the power i+j attached, which is the familiar checkerboard of pluses and minuses starting with a plus in the top-left corner. The cofactor matrix collects all n squared of those. The adjugate is that matrix transposed — and the transpose is not optional decoration, it is what makes the defining identity work. This page proves it rather than claiming it: it multiplies A by adj(A) at runtime and reports the result, which comes out as the determinant times the identity matrix every time.

That identity is the whole point of the adjugate. Because A times adj(A) equals det(A) times I, dividing through gives the inverse: A inverse = adj(A) divided by det(A), which is exactly the formula MIT's 18.06 course states as A inverse equals one over det A times C transpose. When the determinant is zero the division is impossible, and the calculator says so explicitly — the adjugate still exists and A times adj(A) is simply the zero matrix, but no inverse does.

One warning that belongs beside the result rather than in a footnote. 'Adjoint' has two entirely different meanings in linear algebra. This page computes the classical adjoint, the adjugate, the transpose of the cofactor matrix. The other meaning is the Hermitian adjoint A*, the conjugate transpose, which for a real matrix is just A transposed and has nothing to do with minors. If you came here from a functional analysis or quantum mechanics context, the transpose calculator is what you want, not this one.

A practical note on scale: the adjugate route to an inverse is elegant and completely impractical beyond about 3 by 3, because computing n squared minors of size n minus 1 grows explosively. Real software uses an LU factorisation. The adjugate remains valuable for exactly the reason it is taught — it gives a closed-form expression for the inverse, and closed forms are what proofs are made of.

What is cofactor matrix calculator?

For an n x n matrix A, the minor M(i,j) is the determinant of the (n-1) x (n-1) matrix obtained by deleting row i and column j. The cofactor C(i,j) is the signed minor, C(i,j) = (-1)^(i+j) times M(i,j), where the sign follows a checkerboard pattern beginning with a plus in the top-left position. The cofactor matrix C collects every C(i,j) in its natural position. The adjugate, written adj(A) and also called the classical adjoint, is the transpose of the cofactor matrix: adj(A)(i,j) = C(j,i). Its defining property is A times adj(A) = adj(A) times A = det(A) times the identity matrix, which follows from the Laplace expansion in its full form — the Encyclopedia of Mathematics states it as the sum over j of a(i,j) times A(k,j) equals delta(i,k) times det A, meaning a row expanded against its own cofactors gives the determinant while a row expanded against a different row's cofactors gives zero. Dividing that identity by det(A) yields the closed-form inverse A inverse = adj(A) / det(A), valid whenever det(A) is not zero. Two conventions worth stating: for a 1 x 1 matrix the empty determinant is 1, so C = adj = [[1]]; and the classical adjoint should not be confused with the Hermitian adjoint A*, the conjugate transpose, which is a different object entirely.

How to use this calculator.

1. 01Type your square matrix into the box, one row per line, with entries separated by commas or spaces.
2. 02Read the cofactor matrix in the headline. Each entry is the minor with its checkerboard sign attached.
3. 03Read the adjugate below it — that is adj(A), the classical adjoint, and it is simply the cofactor matrix transposed.
4. 04Use the matrix of minors if you want to see the unsigned intermediate step, which is how the calculation is usually taught.
5. 05Check the verification line: it multiplies A by adj(A) and confirms the product is det(A) times the identity.
6. 06To get the inverse, divide every entry of the adjugate by the determinant — but only if the determinant is not zero.
7. 07If you wanted the conjugate transpose rather than the classical adjoint, use the transpose calculator instead; the two share a name and nothing else.

The formula.

Start with the minors. M(i,j) is the determinant of the smaller matrix left after deleting row i and column j, so for A = [[1,2,3],[0,4,5],[1,0,6]] the minor M11 is the determinant of [[4,5],[0,6]] = 24, M12 is the determinant of [[0,5],[1,6]] = -5, and so on down to M33 = the determinant of [[1,2],[0,4]] = 4. Collecting all nine gives the matrix of minors [[24,-5,-4],[12,3,-2],[-2,5,4]]. Next apply the checkerboard sign (-1) to the power i+j, which flips the sign in every position where the row and column indices have opposite parity, turning that into the cofactor matrix C = [[24,5,-4],[-12,3,2],[-2,-5,4]]. Finally transpose it: adj(A) = C transposed = [[24,-12,-2],[5,3,-5],[-4,2,4]]. The determinant falls out for free by expanding the first row against its own cofactors: det = 1 times 24 + 2 times 5 + 3 times -4 = 24 + 10 - 12 = 22. The transpose step is the one people skip, and it is silent for a symmetric matrix, which is why the example here is deliberately not symmetric. To see that it matters, multiply the first row of A by the second column of adj(A): 1 times -12 + 2 times 3 + 3 times 2 = -12 + 6 + 6 = 0, exactly as the identity A times adj(A) = det(A) times I requires. Had the transpose been left out, that dot product would have been 1 times 5 + 2 times 3 + 3 times -5 = -4, and the identity would have failed. Every minor here is computed by a division-free Laplace expansion at 80 significant digits, so an integer matrix produces exact integer minors, cofactors and adjugate entries.

A worked example.

Frequently asked questions.

References& sources.

1. [1]Encyclopedia of Mathematics (EMS Press), 'Determinant'. States the Laplace expansion in its full form, sum_j a_ij A_kj = delta_ik det A, where A_ij is 'the cofactor of entry a_ij' — the identity that is equivalent to A times adj(A) = det(A) times I. Retrieved 2026-07-29.
2. [2]MIT OpenCourseWare 18.06, Lecture 20 — 'Cramer's rule, inverse matrix, and volume' (Strang). States that A^-1 = (1/det A) C^T, where C is the matrix of cofactors of A: the adjugate is the transpose of the cofactor matrix. Retrieved 2026-07-29.
3. [3]Golub, G. H. & Van Loan, C. F. (2013). Matrix Computations, 4th ed., Johns Hopkins University Press, section 3.2 — including why the adjugate route to an inverse is never the right numerical choice beyond very small matrices. Print reference; publisher page linked.