What is cofactor expansion?

Cofactor expansion, also called Laplace expansion, is a method for computing the determinant of a square matrix by expanding along a chosen row or column. For each element in that row or column, you multiply the element by its cofactor (the signed minor formed by deleting that element's row and column). The determinant equals the sum of these products. For a 3x3 matrix expanded along row 1: det(A) = a11*C11 + a12*C12 + a13*C13, where each cofactor Cij = (-1)^(i+j) * Mij and Mij is the 2x2 minor determinant.

Does it matter which row or column I expand along?

No, the result is the same regardless of which row or column you choose for the expansion. The determinant is a property of the matrix, not of the expansion path. Choosing a row or column that contains zeros simplifies the calculation, since any term with a zero element contributes nothing to the sum. For example, if a row has two zeros, only one minor needs to be computed.

What is the difference between a minor and a cofactor?

The minor M_ij is the determinant of the submatrix formed by deleting row i and column j from the original matrix. The cofactor C_ij is the signed minor: C_ij = (-1)^(i+j) * M_ij. The sign factor (-1)^(i+j) produces a checkerboard pattern of plus and minus signs across the matrix. For a 3x3 matrix the sign pattern is: + - + / - + - / + - +. The cofactor matrix (also called the matrix of cofactors) is used in computing the adjugate (classical adjoint) and the matrix inverse.

Cofactor Expansion Calculator

This calculator finds the determinant of a 2x2 or 3x3 matrix using cofactor expansion, also called Laplace expansion, a standard linear algebra method for matrices too large for simple cross-multiplication. Determinants matter because they show whether a matrix is invertible, are needed to solve systems of equations, and underpin techniques such as finding the adjugate and the matrix inverse. To use it, choose your matrix size, pick which row or column to expand along (the answer is identical whichever you choose), and enter the values into the matrix grid. The calculator multiplies each element in your chosen row or column by its cofactor, the signed minor formed by deleting that element's row and column, and sums the results. Your results show the determinant itself, whether the matrix is singular or invertible (a zero determinant means singular, with no inverse), and how many non-zero terms contributed to the sum. Below the results you get full step-by-step working showing each element, its sign, minor and cofactor, plus a complete cofactor table for the whole matrix. This makes it useful for checking homework, verifying hand calculations, or building intuition for how expansion by minors works before you tackle larger matrices with methods such as row reduction. Choosing a row or column with zeros speeds up the working, since a zero element contributes nothing to the total.

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Standard method Laplace expansion by minors and cofactors. Valid for any square matrix.

1. Matrix Size and Expansion

Matrix size

Expand along

2. Enter Matrix Values

Result

Determinant
22
det(A)

Matrix is

Invertible

det = 0 means singular

Expansion terms

3 of 3

Non-zero contributions

Step-by-Step Working

Enter matrix values above to see working.

Cofactor Table

Enter matrix values above to see the cofactor table.

How Cofactor Expansion Works

Cofactor expansion (also called Laplace expansion) is a recursive method for computing the determinant of a square matrix. You pick any single row or column, then express the determinant as a weighted sum of smaller determinants (minors) from the remaining rows and columns.

The Formula

For an n x n matrix A, expanding along row i:

det(A) = a_i1 C_i1 + a_i2 C_i2 + ... + a_in C_in

where the cofactor C_ij = (-1)^i+j M_ij and M_ij is the minor (the determinant of the submatrix obtained by deleting row i and column j).

The sign pattern (-1)^i+j produces a checkerboard of plus and minus across the matrix:

Position	2x2 signs	3x3 signs
Row 1	+ -	+ - +
Row 2	- +	- + -
Row 3	-	+ - +

2x2 Determinant

For a 2x2 matrix [a b; c d], the determinant is simply ad - bc. Cofactor expansion along row 1 gives: det = a * (+1) * d + b * (-1) * c = ad - bc.

3x3 Worked Example (Default Values)

For the matrix A with row 1 = [1, 2, 3], row 2 = [0, 4, 5], row 3 = [1, 0, 6], expanding along row 1:

a₁₁ = 1, sign = +1, minor M₁₁ = det[4,5;0,6] = 24 - 0 = 24, term = 1 * 24 = 24
a₁₂ = 2, sign = -1, minor M₁₂ = det[0,5;1,6] = 0 - 5 = -5, term = 2 * -1 * (-5) = 10
a₁₃ = 3, sign = +1, minor M₁₃ = det[0,4;1,0] = 0 - 4 = -4, term = 3 * (-4) = -12
det(A) = 24 + 10 + (-12) = 22

Choosing the Best Row or Column

Any row or column gives the same answer. Picking the row or column with the most zeros reduces the number of minor computations needed. For example, if row 2 contains two zeros, only one 2x2 minor needs to be calculated. This is why cofactor expansion is particularly efficient when a matrix is already in a sparse or triangular form.

Relationship to Matrix Inverse

The cofactor matrix (where each entry is replaced by its cofactor C_ij) is the foundation of the adjugate matrix (transpose of the cofactor matrix). The matrix inverse is then: A^-1 = (1/det(A)) * adj(A). This relationship is only valid when det(A) is not zero. A matrix with det = 0 is singular and has no inverse.

Related Calculators

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Matrix Determinant Calculator: general determinant finder for 2x2, 3x3, and larger matrices.
3x3 Determinant Calculator: dedicated 3x3 determinant tool.
Matrix Inverse Calculator: find the inverse of a square matrix.
Adjoint Matrix Calculator: compute the classical adjoint (adjugate) from cofactors.

Method: Laplace expansion by cofactors. For each element in the chosen row or column, the cofactor is C_ij = (-1)^i+j * det(M_ij), where M_ij is the (n-1) x (n-1) minor submatrix. The determinant equals the dot product of the selected row/column with its cofactors. Reference: Anton, H. and Rorres, C., Elementary Linear Algebra; Strang, G., Introduction to Linear Algebra.

This calculator supports 2x2 and 3x3 matrices. All entries are treated as real numbers. Results may show floating-point rounding for non-integer inputs; if your inputs are integers the determinant will always be an exact integer.