Adjoint (Adjugate) Matrix Calculator
Enter your square matrix, choose its size (2x2, 3x3, or 4x4), and this calculator finds the adjoint (also called the adjugate or classical adjoint) instantly. It also displays the cofactor matrix, the determinant, and the inverse matrix when it exists. Every step of the working is shown so you can follow the maths and check your own calculations.
Formula
Worked example
For A = [[1,2,3],[0,4,5],[1,0,6]]: C_11 = det([[4,5],[0,6]]) = 24, C_12 = -det([[0,5],[1,6]]) = 5, C_13 = det([[0,4],[1,0]]) = -4. Completing all 9 cofactors and transposing: adj(A) = [[24,-12,-2],[-5,3,-5],[-4,2,4]]. det(A) = 1*24 + 2*(-5) + 3*(-4) = 2. So A-inverse = adj(A)/2 = [[12,-6,-1],[-2.5,1.5,-2.5],[-2,1,2]].
What is the adjoint (adjugate) of a matrix?
The adjoint of a square matrix A, written adj(A), is the transpose of the cofactor matrix of A. It is also called the adjugate or the classical adjoint. Each entry of the cofactor matrix is a signed minor: the cofactor C_ij equals (-1)^(i+j) times the determinant of the (n-1) x (n-1) submatrix you obtain by removing row i and column j from A. Arranging all n^2 cofactors into a matrix and then transposing (swapping rows with columns) gives adj(A). The central property is the identity A * adj(A) = det(A) * I: multiplying a matrix by its adjoint always yields a scalar multiple of the identity matrix.
How to compute the adjoint step by step
The calculation has three clear stages. First, compute the minor M_ij for each position (i, j): remove row i and column j from A and find the determinant of the smaller matrix that remains. Second, apply the sign factor (-1)^(i+j) to each minor to form the cofactor C_ij. The signs follow a checkerboard pattern: plus on the main diagonal and at positions where i+j is even, minus where i+j is odd. Third, transpose the cofactor matrix by moving the entry at position (i, j) to position (j, i). The result is adj(A). For a 2x2 matrix the process reduces to a simple rule: swap the two main-diagonal entries and negate the two off-diagonal entries.
The adjoint and the matrix inverse
When the determinant of A is non-zero, the inverse is A-inverse = adj(A) / det(A). Each entry of adj(A) is divided by the scalar det(A). When det(A) equals zero the matrix is singular, it has linearly dependent rows or columns, and no inverse exists - though the adjoint itself is still well-defined. In practice, Gaussian elimination is faster for solving large systems, but the adjoint formula gives a closed-form symbolic answer. It is also the foundation of Cramer's rule for solving Ax = b: each unknown x_i equals det(A with column i replaced by b) divided by det(A), which can be derived from the adjoint formula.
Adjoint vs Hermitian adjoint: a naming caution
The word "adjoint" appears in two different mathematical contexts. In linear algebra the adjoint (this calculator) is the adjugate: transpose of the cofactor matrix, computed from determinants. In functional analysis and quantum mechanics "adjoint" usually means the Hermitian adjoint or conjugate transpose, written A-dagger, where every entry is replaced by its complex conjugate before transposing. These are entirely different operations. If you are working with real matrices the two concepts coincide only for orthogonal matrices (where A-inverse = A-transpose). For complex matrices they are different. This calculator computes the linear-algebra adjugate only.
Sign pattern (+/-) for cofactors in a 3x3 matrix
| Position | i+j | Sign | Cofactor formula |
|---|---|---|---|
| (1,1) | 2 | + | C_11 = +M_11 |
| (1,2) | 3 | - | C_12 = -M_12 |
| (1,3) | 4 | + | C_13 = +M_13 |
| (2,1) | 3 | - | C_21 = -M_21 |
| (2,2) | 4 | + | C_22 = +M_22 |
| (2,3) | 5 | - | C_23 = -M_23 |
| (3,1) | 4 | + | C_31 = +M_31 |
| (3,2) | 5 | - | C_32 = -M_32 |
| (3,3) | 6 | + | C_33 = +M_33 |
Cofactor sign = (-1)^(i+j). Even sum of indices gets +, odd sum gets -. This checkerboard pattern applies to any size matrix.
Frequently asked questions
What is the difference between the adjoint and the adjugate?
They are the same object. "Adjugate" is the preferred modern name, used to avoid confusion with the Hermitian adjoint (conjugate transpose) from functional analysis. Classical textbooks call it the "adjoint" or "classical adjoint." This calculator uses both names interchangeably because both appear in textbooks and course materials.
How do I find the adjoint of a 2x2 matrix quickly?
For a 2x2 matrix [[a, b], [c, d]], the adjoint is [[d, -b], [-c, a]]: swap the two main-diagonal entries and negate the two off-diagonal entries. No submatrix determinant is needed. This shortcut follows from the fact that the cofactor of each entry in a 2x2 matrix is just a single element of the other matrix with a possible sign flip.
Can I use the adjoint to find the matrix inverse?
Yes: A-inverse = adj(A) / det(A), provided det(A) is non-zero. Divide every entry of adj(A) by the determinant. If det(A) = 0 the matrix is singular and has no inverse. The adjoint formula is exact and symbolic, which is why it appears in proofs and for small matrices where exact rational arithmetic matters, even though row reduction is more efficient for large numerical matrices.
What does a zero determinant mean for the adjoint?
A zero determinant means the matrix is singular: at least one row is a linear combination of others. The adjoint adj(A) still exists and can be computed, but A-inverse does not exist. The identity A * adj(A) = det(A) * I becomes A * adj(A) = 0, the zero matrix. The system Ax = b either has no solution or infinitely many solutions when det(A) = 0.
Why does the sign alternate in a checkerboard pattern for cofactors?
The sign factor (-1)^(i+j) is required so that cofactor expansion gives the correct determinant along any row or column. When you expand det(A) along row i, each term is A[i][j] times C_ij, and the alternating signs ensure the Laplace expansion formula is consistent: you can expand along any row or column and always get the same determinant. Without the sign pattern the expansion would not be well-defined.
Does the adjoint work for non-square matrices?
No. The adjugate is only defined for square matrices because it relies on cofactors and determinants, which require square sub-matrices. For non-square (rectangular) matrices the relevant generalisation is the Moore-Penrose pseudoinverse, computed via singular value decomposition rather than cofactors.
How is the adjoint related to Cramer's rule?
Cramer's rule expresses the solution of Ax = b as x_i = det(A_i) / det(A), where A_i is A with column i replaced by b. This follows directly from the adjoint: x = A-inverse * b = (adj(A) / det(A)) * b, and expanding that product entry by entry gives exactly Cramer's rule. So the adjoint provides the algebraic underpinning of Cramer's rule.