Cofactor Matrix Calculator
The cofactor matrix is built from the signed minors of a matrix and is the bridge to both the determinant and the inverse. Each entry is the determinant of the 2x2 block left after deleting its row and column, multiplied by a checkerboard sign. This calculator takes the nine entries of a 3x3 matrix, computes all nine cofactors, and also reports the determinant by first-row cofactor expansion. Transposing the cofactor matrix yields the adjugate, the next step toward the inverse.
Cofactor formula
C(i,j) = (-1)^(i+j) * M(i,j)
M(i,j) = det of 2x2 minor after deleting
row i and column j
sign pattern: + - + / - + - / + - +
det = a11*C11 + a12*C12 + a13*C13
Each cofactor combines a 2x2 minor determinant with a checkerboard sign. Expanding the first row against its cofactors recovers the full 3x3 determinant.
Notes on cofactors
- Each input row must contain exactly three comma-separated numbers.
- The sign of each cofactor follows the (-1) to the (i+j) checkerboard.
- The transpose of the cofactor matrix is the adjugate.
- The inverse is the adjugate divided by the determinant, when the determinant is nonzero.
- First-row cofactor expansion is used here to report the determinant.
Cofactor matrix: frequently asked questions
What is a cofactor matrix?
The cofactor matrix of a square matrix has, in each position, the cofactor of the corresponding entry: the determinant of the 2x2 minor left after deleting that entry's row and column, multiplied by a sign of plus or minus one based on position.
How is each cofactor calculated?
The cofactor C(i,j) equals (-1) raised to the (i+j) power times the minor M(i,j), where the minor is the determinant of the 2x2 submatrix obtained by deleting row i and column j. The sign pattern alternates like a checkerboard.
How does the cofactor matrix relate to the determinant?
Expanding any row or column against its cofactors gives the determinant. This calculator uses the first row: the determinant equals a11 times C11 plus a12 times C12 plus a13 times C13.
How does the cofactor matrix relate to the inverse?
Transposing the cofactor matrix gives the adjugate, and dividing the adjugate by the determinant gives the inverse. So the cofactor matrix is the first step in computing a matrix inverse by the classical adjugate method.
What is the sign pattern of the cofactors?
The signs follow a checkerboard: plus, minus, plus across the first row; minus, plus, minus across the second; plus, minus, plus across the third. This is the value of (-1) to the (i+j) power for each position.
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.