Calculate matrix determinants and cofactors using cofactor expansion.
Last updated: 2026-05-24T22:58:31.706Z
Matrix entries must be numeric. Use decimal point for non-integers.
Determinant
5.000000
Cofactor Matrix
Cofactor expansion (also called Laplace expansion) is a method for calculating the determinant of a matrix by breaking it down into smaller matrices. This recursive approach is particularly useful for understanding the structure of determinants and working with larger matrices.
A cofactor is a signed minor of a matrix. The sign alternates in a checkerboard pattern (positive for even row+column sum, negative for odd). Cofactors are essential in finding inverse matrices, solving systems of equations, and understanding matrix transformations.
Direct calculation gives the determinant immediately.
Expand along the first row, multiplying each element by its cofactor.
Where Mᵢⱼ is the minor (determinant of the submatrix)
Calculate the determinant using cofactor expansion:
Matrix: [[2, 3], [1, 4]]
Step 1: Apply formula: det = a·d - b·c
Step 2: Substitute: det = (2)(4) - (3)(1)
Step 3: Calculate: det = 8 - 3 = 5
Result: Determinant = 5
A minor is the determinant of a smaller matrix obtained by deleting one row and one column from the original matrix.
A cofactor is a signed minor. The sign depends on the position: (-1)^(i+j) where i and j are row and column indices.
Yes. Expanding along any row or column gives the same determinant, so choose the row/column with the most zeros for efficiency.
It breaks down complex determinant calculations into simpler ones, and it helps understand matrix properties and structure.
The inverse matrix uses the adjugate matrix (transpose of cofactor matrix) divided by the determinant.
A zero determinant means the matrix is singular (non-invertible) and its rows/columns are linearly dependent.
Yes, but computationally inefficient. For 4×4 or larger, methods like Gaussian elimination are preferred.
Determinants solve systems of equations (Cramer's rule), compute areas/volumes, and determine linear independence.
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