Calculate the matrix of cofactors from any 2×2 or 3×3 matrix.
Last updated: 2026-05-24T22:58:31.711Z
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Cofactor Matrix
A cofactor matrix is obtained by replacing each element of a matrix with its corresponding cofactor. Each cofactor is a signed minor (the determinant of the smaller matrix formed by deleting a row and column, multiplied by -1 raised to the sum of the row and column indices).
The cofactor matrix is essential for finding the inverse matrix and understanding matrix transformations. The transpose of the cofactor matrix is called the adjugate matrix, which is used in the formula: A⁻¹ = (1/det(A)) × adj(A).
Where Mᵢⱼ is the minor determinant
Where adj(A) is transpose of cofactor matrix
Calculate the cofactor matrix for [[2, 3], [1, 4]]:
Step 1: C₁₁ = (-1)^(1+1) · 4 = 4
Step 2: C₁₂ = (-1)^(1+2) · 1 = -1
Step 3: C₂₁ = (-1)^(2+1) · 3 = -3
Step 4: C₂₂ = (-1)^(2+2) · 2 = 2
Result: Cofactor Matrix = [[4, -1], [-3, 2]]
A minor is the determinant of a submatrix. A cofactor is a signed minor with the sign (-1)^(row+col).
Cofactors are essential for computing matrix inverses, solving systems of equations, and understanding determinants.
Delete the row i and column j, then find the determinant of the remaining smaller matrix.
The adjugate matrix is the transpose of the cofactor matrix. It's used to calculate matrix inverses.
Yes, the cofactor method works for any square matrix, though it becomes computationally expensive for large matrices.
If det(A) = 0, the matrix is singular and non-invertible. The cofactor matrix still exists but can't be used for inversion.
Cramer's rule uses determinants and cofactors to solve systems of linear equations efficiently.
Yes, cofactors are used in computer graphics, engineering simulations, and numerical analysis.
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