Expansion by Cofactors

Expansion by Cofactors

A method for evaluating determinants. Expansion by cofactors involves following any row or column of a determinant and multiplying each element of the row or column by its cofactor. The sum of these products equals the value of the determinant.

3x3 determinant expanded along column 3: 3|4,5;7,8| -6|1,2;7,8| +9|1,2;4,5| = 3(-3)-6(-6)+9(-3) = 0

Key Formula

\det(A) = \sum_{j=1}^{n} a_{ij} \, C_{ij} \quad \text{(expansion along row } i\text{)}

Where:

$a_{ij}$ = The element in row i and column j of the matrix A
$C_{ij}$ = The cofactor of element a_{ij}, defined as C_{ij} = (-1)^{i+j} M_{ij}
$M_{ij}$ = The minor of a_{ij}, which is the determinant of the submatrix formed by deleting row i and column j
$n$ = The number of rows (or columns) of the square matrix

Worked Example

Problem: Find the determinant of the 3×3 matrix A by expanding along the first row: A = [[2, 1, 3], [0, 4, 5], [1, 0, 2]]

Step 1: Write the cofactor expansion formula along row 1.

\det(A) = a_{11}C_{11} + a_{12}C_{12} + a_{13}C_{13}

Step 2: Compute cofactor C₁₁. Delete row 1 and column 1, then find the 2×2 determinant. The sign factor is (-1)^(1+1) = +1.

C_{11} = (+1)\begin{vmatrix} 4 & 5 \\ 0 & 2 \end{vmatrix} = (4)(2) - (5)(0) = 8

Step 3: Compute cofactor C₁₂. Delete row 1 and column 2. The sign factor is (-1)^(1+2) = -1.

C_{12} = (-1)\begin{vmatrix} 0 & 5 \\ 1 & 2 \end{vmatrix} = -(0 \cdot 2 - 5 \cdot 1) = -(-5) = 5

Step 4: Compute cofactor C₁₃. Delete row 1 and column 3. The sign factor is (-1)^(1+3) = +1.

C_{13} = (+1)\begin{vmatrix} 0 & 4 \\ 1 & 0 \end{vmatrix} = (0)(0) - (4)(1) = -4

Step 5: Substitute and sum.

\det(A) = 2(8) + 1(5) + 3(-4) = 16 + 5 - 12 = 9

Answer: det(A) = 9

Another Example

This example shows that you can expand along any row or column and get the same determinant. Choosing a row or column with zeros reduces the number of cofactors you need to compute.

Problem: Find the determinant of the same matrix A = [[2, 1, 3], [0, 4, 5], [1, 0, 2]] by expanding along column 2 instead of row 1.

Step 1: Write the cofactor expansion along column 2. Notice that a₃₂ = 0, so its term vanishes, saving work.

\det(A) = a_{12}C_{12} + a_{22}C_{22} + a_{32}C_{32}

Step 2: We already found C₁₂ = 5 in the previous example. Now compute C₂₂. Delete row 2 and column 2; sign factor is (-1)^(2+2) = +1.

C_{22} = (+1)\begin{vmatrix} 2 & 3 \\ 1 & 2 \end{vmatrix} = (2)(2) - (3)(1) = 1

Step 3: Since a₃₂ = 0, the third term is zero regardless of the cofactor value.

a_{32}C_{32} = 0 \cdot C_{32} = 0

Step 4: Substitute and sum.

\det(A) = 1(5) + 4(1) + 0 = 5 + 4 = 9

Answer: det(A) = 9, confirming the result from expanding along row 1.

Frequently Asked Questions

Does it matter which row or column you expand along?

No. You will always get the same determinant value regardless of which row or column you choose. However, picking a row or column that contains zeros makes the calculation easier, because any term with a zero element contributes nothing to the sum.

What is the difference between a minor and a cofactor?

The minor M_{ij} is the determinant of the smaller matrix obtained by deleting row i and column j. The cofactor C_{ij} equals the minor multiplied by a sign factor: C_{ij} = (-1)^{i+j} M_{ij}. The sign alternates in a checkerboard pattern starting with + in the top-left corner.

When should you use cofactor expansion instead of row reduction?

Cofactor expansion is most practical for 2×2 and 3×3 matrices or for matrices that contain many zeros. For larger matrices (4×4 and above), row reduction to triangular form is usually faster because cofactor expansion requires computing many sub-determinants.

Cofactor Expansion vs. Row Reduction (for determinants)

	Cofactor Expansion	Row Reduction (for determinants)
Method	Multiply elements by cofactors and sum	Use row operations to reach triangular form, then multiply diagonal entries
Best for	Small matrices (2×2, 3×3) or matrices with many zeros	Larger matrices (4×4 and above)
Computational cost	Grows very fast (roughly n! operations for an n×n matrix)	Grows more slowly (roughly n³ operations)
Gives symbolic formulas?	Yes — produces an explicit algebraic expression	Less convenient for symbolic work

Why It Matters

Cofactor expansion appears in nearly every linear algebra course as the standard way to compute determinants by hand for 3×3 and small matrices. It also underlies the formula for the inverse of a matrix using the adjugate (matrix of cofactors), which connects determinants to systems of equations via Cramer's rule. Understanding this method builds intuition for how each element of a matrix contributes to the determinant's value.

Common Mistakes

Mistake: Forgetting the sign pattern. Students often multiply every element by its minor without the (-1)^{i+j} factor.

Correction: Always apply the checkerboard sign pattern: +, −, +, −, ... starting from (+1) for the element in position (1,1). A quick way to remember is that the sign is positive when i + j is even and negative when i + j is odd.

Mistake: Deleting the wrong row or column when forming the minor.

Correction: For element a_{ij}, you must delete row i AND column j — the row and column that the element sits in. Double-check by verifying your submatrix has dimensions (n−1) × (n−1).

Related Terms

Determinant — The value computed by cofactor expansion
Cofactor — Signed minor used in each expansion term
Element of a Matrix — Individual entry multiplied by its cofactor
Row of a Matrix — A row can be chosen for the expansion
Column of a Matrix — A column can be chosen for the expansion
Evaluate — Cofactor expansion evaluates a determinant
Sum — Results are summed to get the determinant
Product — Each term is a product of element and cofactor