Cofactor Expansion Calculator

Calculate the determinant of 2x2 and 3x3 matrices using cofactor expansion (Laplace expansion) along any row or column with step-by-step solutions.

Enter Matrix

Enter the elements of the 2x2 matrix:

Results

Determinant
-14
Matrix Size2x2
Expansion MethodDirect formula
Is Singular?No
Is Invertible?Yes

Step-by-Step Cofactor Expansion

det(A) = ad - bc

What is Cofactor Expansion?

Cofactor expansion (also known as Laplace expansion) is a method for computing the determinant of a square matrix by expanding along any row or column. The determinant is expressed as a sum of products of matrix entries and their corresponding cofactors. This method works for matrices of any size, but is most practical for 2x2 and 3x3 matrices.

Key Formulas

2x2 Determinant

For matrix [[a, b], [c, d]], the determinant is computed directly.

det = ad - bc

3x3 Expansion (Row 1)

Expand along the first row using cofactors.

det = a(ei-fh) - b(di-fg) + c(dh-eg)

Cofactor Formula

The cofactor C_ij includes a sign factor.

C_ij = (-1)^(i+j) x M_ij

Minor M_ij

The minor is the determinant of the submatrix obtained by deleting row i and column j.

M_ij = det(submatrix)

How Cofactor Expansion Works

To compute the determinant of an n x n matrix using cofactor expansion:

  1. Choose any row or column to expand along (all give the same result).
  2. For each element in that row or column, compute its minor (the determinant of the submatrix formed by deleting that element's row and column).
  3. Multiply each element by its cofactor: the minor times (-1)^(i+j), where i and j are the row and column indices.
  4. Sum all these products to get the determinant.

Sign Pattern (Checkerboard)

The signs for cofactors follow a checkerboard pattern. For a 3x3 matrix, the sign pattern is:

  • Row 1: +, -, +
  • Row 2: -, +, -
  • Row 3: +, -, +

This pattern is determined by (-1)^(i+j). When i+j is even, the sign is positive; when odd, the sign is negative.

Choosing the Best Row or Column

While you can expand along any row or column and get the same determinant, choosing wisely can simplify calculations significantly. The best strategy is to expand along the row or column with the most zeros, since any term multiplied by zero vanishes and requires no further computation.

Properties of Determinants

  • If the determinant is zero, the matrix is singular (not invertible).
  • Swapping two rows or columns changes the sign of the determinant.
  • Multiplying a row by a scalar k multiplies the determinant by k.
  • Adding a multiple of one row to another does not change the determinant.
  • det(AB) = det(A) x det(B) for square matrices A and B.
  • det(A^T) = det(A), so expanding by rows or columns gives the same result.

Applications of Determinants

  • Linear algebra: Determining if a system of linear equations has a unique solution.
  • Matrix inversion: The inverse of a matrix involves the determinant and the adjugate matrix.
  • Cramer's rule: Solving systems of linear equations using determinants.
  • Geometry: Computing areas (2x2) and volumes (3x3) using determinants.
  • Eigenvalues: The characteristic polynomial involves computing det(A - lambda*I).

Computational Complexity

Cofactor expansion has a computational complexity of O(n!) for an n x n matrix, making it impractical for large matrices. For matrices larger than 4x4, methods like LU decomposition or Gaussian elimination are preferred, offering O(n^3) complexity. However, cofactor expansion remains valuable for small matrices and for understanding the theoretical foundations of linear algebra.