Determinant Expansion by Minors — Definition, Formula & Examples

Determinant expansion by minors is a method for computing the determinant of a square matrix by multiplying each entry in a chosen row or column by its cofactor, then summing the results.

For an $n \times n$ matrix $A$ , the determinant can be computed by expanding along row $i$ as $\det(A) = \sum_{j=1}^{n} a_{ij}\,C_{ij}$ , where $a_{ij}$ is the $(i,j)$ -entry and $C_{ij} = (-1)^{i+j} M_{ij}$ is the cofactor, with $M_{ij}$ denoting the minor obtained by deleting row $i$ and column $j$ . An analogous formula holds for expansion along any column.

Key Formula

\det(A) = \sum_{j=1}^{n} a_{ij}\,(-1)^{i+j}\,M_{ij}

Where:

$a_{ij}$ = Entry in row i, column j of matrix A
$M_{ij}$ = Minor: determinant of the submatrix formed by deleting row i and column j
$(-1)^{i+j}$ = Sign factor that alternates in a checkerboard pattern
$n$ = Size of the square matrix

How It Works

Pick any row or column of the matrix — the result is the same regardless of your choice. For each entry in that row or column, delete the entry's row and column to form a smaller submatrix; the determinant of that submatrix is the minor. Multiply the entry by its minor and by the sign factor

(-1)^{i+j}

, which alternates in a checkerboard pattern starting with

+

at position

(1,1)

. Sum all these signed products to get the determinant. Choosing a row or column with zeros reduces computation, since those terms vanish.

Worked Example

Problem: Find the determinant of the 3×3 matrix A by expanding along the first row, where A = [[2, 0, 1], [3, 1, 4], [5, 2, 6]].

Step 1: Identify entries and sign factors for row 1:

a_{11}=2

with sign

+

a_{12}=0

with sign

-

a_{13}=1

with sign

+

Step 2: Compute the minor for

a_{11}=2

by deleting row 1 and column 1.

M_{11} = \det\begin{pmatrix} 1 & 4 \\ 2 & 6 \end{pmatrix} = (1)(6)-(4)(2) = -2

Step 3: Skip

a_{12}=0

since its contribution is zero. Compute the minor for

a_{13}=1

M_{13} = \det\begin{pmatrix} 3 & 1 \\ 5 & 2 \end{pmatrix} = (3)(2)-(1)(5) = 1

Step 4: Combine the terms using the cofactor expansion formula.

\det(A) = (+1)(2)(-2) + 0 + (+1)(1)(1) = -4 + 0 + 1 = -3

Answer:

\det(A) = -3

Why It Matters

This technique is essential for deriving Cramer's Rule, computing the classical adjoint, and understanding theoretical properties of determinants. In courses on linear algebra and differential equations, cofactor expansion is often the first determinant method students learn before moving to row reduction approaches for larger matrices.

Common Mistakes

Mistake: Using the wrong sign factor — for example, applying

+

to every term instead of alternating with

(-1)^{i+j}

Correction: Always use the checkerboard sign pattern. For row 1: the signs go

+, -, +, -, \ldots

For row 2:

-, +, -, +, \ldots

and so on.