01 / 21

Cofactor Expansion
& Minors

Master the technique of computing determinants by expanding along rows or columns using minors and cofactors.

Linear Algebra

What Is a Determinant?

Every square matrix \( A \) has a single number called its determinant, written \( \det(A) \) or \( |A| \). It tells us whether the matrix is invertible, how it scales area or volume, and whether its system of equations has a unique solution.

\( \det(A) \neq 0 \)
Matrix is invertible: unique solution exists
\( \det(A) = 0 \)
Matrix is singular: no inverse, dependent rows

Understanding Minors

The minor \( M_{ij} \) of entry \( a_{ij} \) is the determinant of the smaller matrix formed by deleting row \( i \) and column \( j \) from the original matrix.

Delete row 1 & col 2 Remaining 2×2 submatrix
\(a_{11}\)
\(a_{12}\)
\(a_{13}\)
\(a_{21}\)
\(a_{22}\)
\(a_{23}\)
\(a_{31}\)
\(a_{32}\)
\(a_{33}\)
\(a_{21}\)
\(a_{23}\)
\(a_{31}\)
\(a_{33}\)
Minor \(M_{12}\)

From Minors to Cofactors

The cofactor \( C_{ij} \) attaches a sign to each minor. The sign alternates in a checkerboard pattern determined by \( (-1)^{i+j} \).

\( C_{ij} = (-1)^{i+j} \cdot M_{ij} \)
Sign Pattern (3×3)
\(+\)
\(-\)
\(+\)
\(-\)
\(+\)
\(-\)
\(+\)
\(-\)
\(+\)

The Cofactor Expansion Theorem

The determinant of an \( n \times n \) matrix can be computed by expanding along any single row or column. Each entry is multiplied by its cofactor and summed.

Expansion along row \( i \):
\( \det(A) = \displaystyle\sum_{j=1}^{n} a_{ij} \cdot C_{ij} = \sum_{j=1}^{n} a_{ij} \cdot (-1)^{i+j} \cdot M_{ij} \)
Expansion along column \( j \):
\( \det(A) = \displaystyle\sum_{i=1}^{n} a_{ij} \cdot C_{ij} = \sum_{i=1}^{n} a_{ij} \cdot (-1)^{i+j} \cdot M_{ij} \)

Why Does This Work?

Cofactor expansion is a recursive decomposition. Each \( n \times n \) determinant is broken into \( n \) smaller \( (n-1) \times (n-1) \) determinants, which are themselves broken down, until we reach \( 1 \times 1 \) matrices.

Sum all three terms to get det(A)
\(3 \times 3\) det
\(2 \times 2\) det
\(2 \times 2\) det
\(2 \times 2\) det
\(+a_{11} \times\)
\(-a_{12} \times\)
\(+a_{13} \times\)
\(ad - bc\)
\(ad - bc\)
\(ad - bc\)

Minor vs Cofactor

Minor \( M_{ij} \) Cofactor \( C_{ij} \)
Definition Determinant of submatrix after deleting row \( i \), col \( j \) Signed minor: \( (-1)^{i+j} M_{ij} \)
Sign Can be positive, negative, or zero (it is a determinant) Includes the \( (-1)^{i+j} \) factor: alternates \( +, -, +, -, \ldots \)
Example: \( M_{12} \) \( M_{12} = 5 \) \( C_{12} = (-1)^{1+2} \cdot 5 = -5 \)
Used in Intermediate calculation Determinant formula, inverse matrix

How to Apply Cofactor Expansion

1
Choose a row or column. Pick the one with the most zeros to reduce work.
2
For each entry in that row/column, compute the sign \( (-1)^{i+j} \).
3
Find the minor by deleting the entry's row and column and computing the remaining determinant.
4
Multiply each entry by its sign and its minor (this gives the cofactor term).
5
Sum all terms to get the determinant.

Example 1: The \( 2 \times 2 \) Case

Let's verify cofactor expansion on a simple \( 2 \times 2 \) matrix where we already know the formula \( ad - bc \).

\( A = \begin{pmatrix} 3 & 7 \\ 2 & 5 \end{pmatrix} \)

Example 2: A \( 3 \times 3 \) Determinant

\( A = \begin{pmatrix} 1 & 2 & 3 \\ 0 & 4 & 5 \\ 1 & 0 & 6 \end{pmatrix} \)

Strategy: Expand along column 1 because it has a zero at position \( (2,1) \), which eliminates one term entirely.

Example 2: Verification

Let's verify by expanding along row 2 instead (which also has a zero).

\( A = \begin{pmatrix} 1 & 2 & 3 \\ 0 & 4 & 5 \\ 1 & 0 & 6 \end{pmatrix} \)

Example 3: A \( 4 \times 4 \) Matrix

\( B = \begin{pmatrix} 2 & 0 & 0 & 1 \\ 3 & 1 & 0 & 0 \\ 0 & 0 & 5 & 0 \\ 4 & 0 & 0 & 2 \end{pmatrix} \)

Strategy: Expand along row 3: it has three zeros, so only one term survives!

Choosing the Best Row or Column

The row or column you expand along determines how much work you do. Each zero entry eliminates an entire minor computation.

Best Case
A row/column with \( n-1 \) zeros: only 1 minor to compute
Good Case
A row/column with some zeros: fewer minors needed
Worst Case
No zeros anywhere: must compute all \( n \) minors

Cofactors and the Inverse Matrix

Cofactors are not just for determinants: they also give us a formula for the matrix inverse. The adjugate (or classical adjoint) is the transpose of the cofactor matrix.

\( A^{-1} = \dfrac{1}{\det(A)} \cdot \text{adj}(A) \)
1
Compute all cofactors \( C_{ij} \) to form the cofactor matrix.
2
Transpose the cofactor matrix to get \( \text{adj}(A) \).
3
Divide by \( \det(A) \). This only works when \( \det(A) \neq 0 \).

Common Mistakes

✕ Wrong Sign Pattern
Forgetting that the sign is \( (-1)^{i+j} \), not always positive. The checkerboard starts with + at position \( (1,1) \).
✕ Deleting Wrong Row/Col
When finding \( M_{ij} \), you must delete row \( i \) and column \( j \): not the row and column of a different entry.
✕ Mixing Up Minor and Cofactor
Using the minor directly in the expansion formula without attaching the \( (-1)^{i+j} \) sign factor.

Summary Table

Concept Symbol Definition
Minor \( M_{ij} \) Det of submatrix (delete row \( i \), col \( j \))
Cofactor \( C_{ij} \) \( (-1)^{i+j} \cdot M_{ij} \)
Determinant \( \det(A) \) \( \sum a_{ij} C_{ij} \) along any row/col
Adjugate \( \text{adj}(A) \) Transpose of cofactor matrix
Inverse \( A^{-1} \) \( \frac{1}{\det(A)} \cdot \text{adj}(A) \)

Question 1 of 5

What is the minor \( M_{12} \) of the matrix \( \begin{pmatrix} 1 & 3 \\ 4 & 2 \end{pmatrix} \)?

Question 2 of 5

What is the cofactor \( C_{12} \) of the matrix \( \begin{pmatrix} 1 & 3 \\ 4 & 2 \end{pmatrix} \)?

Question 3 of 5

Compute \( \det\begin{pmatrix} 2 & 1 & 0 \\ 0 & 3 & 0 \\ 1 & 4 & 5 \end{pmatrix} \) using cofactor expansion along the best row or column.

Question 4 of 5

Which row or column is the most efficient choice for cofactor expansion on the matrix \( \begin{pmatrix} 5 & 0 & 1 \\ 2 & 0 & 7 \\ 1 & 3 & 0 \end{pmatrix} \)?

Question 5 of 5

If a \( 4 \times 4 \) matrix has an entire row of zeros, what is its determinant?