Adjugate Matrix Calculator
The adjugate, or classical adjoint, of a matrix is the transpose of its cofactor matrix and is the key ingredient in the classical formula for the inverse. This calculator takes a 3x3 matrix, computes the nine signed cofactors, transposes them into the adjugate, and reports the determinant alongside. Dividing the adjugate by the determinant yields the inverse whenever the determinant is nonzero. The adjugate also satisfies the elegant identity that a matrix times its adjugate equals the determinant times the identity.
Adjugate formula
C(i,j) = (-1)^(i+j) * minor(i,j)
adjugate = transpose of cofactor matrix
adj(A)[i][j] = C(j,i)
inverse = adjugate / determinant
A * adj(A) = det(A) * I
Each cofactor is a signed 2x2 minor. Transposing the full cofactor matrix gives the adjugate, which divided by the determinant yields the inverse when the determinant is nonzero.
Notes on the adjugate
- Each input row must contain exactly three comma-separated numbers.
- The adjugate is the transpose of the cofactor matrix.
- The inverse equals the adjugate divided by the determinant.
- A matrix times its adjugate equals the determinant times the identity.
- The adjugate exists even when the matrix is singular.
Adjugate matrix: frequently asked questions
What is the adjugate of a matrix?
The adjugate, also called the classical adjoint, is the transpose of the cofactor matrix. Each entry in position (i,j) of the adjugate is the cofactor C(j,i) of the original matrix, with rows and columns swapped.
How is the adjugate used to find an inverse?
The inverse of a matrix equals its adjugate divided by its determinant, provided the determinant is nonzero. This is the classical adjugate formula for matrix inversion and works for any square matrix.
How is the adjugate computed for a 3x3 matrix?
First find the nine cofactors, each a signed 2x2 minor. Arrange them into the cofactor matrix, then transpose it (swap rows and columns). The result is the adjugate.
What is the product of a matrix and its adjugate?
A matrix times its adjugate equals the determinant times the identity matrix. This identity holds even when the determinant is zero, in which case the product is the zero matrix.
Does the adjugate exist for a singular matrix?
Yes. The adjugate is always defined for any square matrix, including singular ones with zero determinant. The inverse, however, only exists when the determinant is nonzero.
Official sources
- NIST Digital Library of Mathematical Functions: Determinants and linear operators.
- NIST/SEMATECH e-Handbook of Statistical Methods: Linear algebra reference.
Reviewed by the CalculatorHub team, edited by James Graham, 16 June 2026. See our methodology.