Cofactor expansion余因子展開よいんしてんかい and testing invertibility可逆性かぎゃくせい

date2026-07-14document_iddoc_948ed6f28f579f32c052b1cacfb93935description余因子展開を置換による行列式の定義から証明し、余因子行列・随伴行列・逆行列公式を通して行列式と可逆性を接続する講義である。prerequisites置換による行列式の定義 / 行列式の計算規則 / 逆行列の基本 / 正方行列type講義content_typelecturestatusactive

mathlinear-algebraundergraduatelecturedeterminant

1Introduction

The key point is that cofactor expansion余因子展開よいんしてんかい is not a new definition of the determinant行列式ぎょうれつしき. It is a theorem derived from the permutation definition. It reduces an $n \times n$ determinant to determinants of size $(n - 1) \times (n - 1)$ .

Cofactors also form the adjugate matrix随伴行列ずいはんぎょうれつ, which proves the inverse formula

A^{- 1} = \frac{1}{\det A} adj (A)

when $\det A \neq 0$ .

2Terms and definitions

Let $A = (a_{i j})$ be an $n \times n$ square matrix正方行列せいほうぎょうれつ. The submatrix小行列しょうぎょうれつ $A^{(i | j)}$ is obtained by deleting row $i$ and column $j$ .

The minor小行列式しょうぎょうれつしき and cofactor余因子よいんし are

M_{i j} = \det A^{(i | j)}, C_{i j} = (- 1)^{i + j} M_{i j} .

The cofactor matrix余因子行列よいんしぎょうれつ is $C (A) = (C_{i j})$ , and the adjugate matrix随伴行列ずいはんぎょうれつ is

adj (A) = C (A)^{T} .

3Plan

To expand along row $i$ , group the permutation sum by the value of $σ (i)$ . The group with $σ (i) = j$ contains the factor $a_{i j}$ , and the remaining choices are exactly the determinant of the submatrix obtained by deleting row $i$ and column $j$ . The sign correction is $(- 1)^{i + j}$ .

4Intuitive explanation

Cofactor expansion decomposes high-dimensional oriented volume by fixing one row or column. The selected entry becomes a coefficient, and the remaining determinant measures the lower-dimensional slice. The checkerboard sign $(- 1)^{i + j}$ corrects orientation.

5Precise explanation

5.11. Cofactor expansion theorem

For an $n \times n$ square matrix $A$ ,

\det A = \sum_{j = 1}^{n} a_{i j} C_{i j}

for every row $i$ , and

\det A = \sum_{i = 1}^{n} a_{i j} C_{i j}

for every column $j$ .

5.22. Proof of row expansion

Start from the permutation definition and group terms by the value of $σ (i)$ . In the group with $σ (i) = j$ , the factor $a_{i j}$ appears. Moving row $i$ and column $j$ to the front requires $(i - 1) + (j - 1)$ adjacent swaps, giving the sign $(- 1)^{i + j}$ . The remaining signed sum is the determinant of $A^{(i | j)}$ . Thus that group contributes $a_{i j} C_{i j}$ , and summing over $j$ proves the row expansion.

Column expansion follows similarly, or by applying row expansion to $A^{T}$ and using $\det A = \det A^{T}$ .

5.33. Adjugate identity

The $(i, k)$ entry of $A adj (A)$ is

\sum_{j = 1}^{n} a_{i j} C_{k j} .

If $i = k$ , this is the cofactor expansion of $\det A$ . If $i \neq k$ , it is the determinant of the matrix obtained by replacing row $k$ by row $i$ , expanded along row $k$ . That matrix has two equal rows, so its determinant is zero. Therefore

A adj (A) = (\det A) I .

The identity $adj (A) A = (\det A) I$ follows similarly from column expansion.

5.44. Inverse formula

If $\det A \neq 0$ , then division by $\det A$ is allowed, so

A^{- 1} = \frac{1}{\det A} adj (A) .

Conversely, if $A$ is invertible, then

\det A \cdot \det (A^{- 1}) = \det I = 1,

so $\det A \neq 0$ . Hence $A$ is invertible if and only if $\det A \neq 0$ .

6Worked example

For

A = (\begin{matrix} 1 & 0 & 2 \\ 3 & 1 & 0 \\ 0 & 4 & 1 \end{matrix}),

expansion along the first row gives

\begin{aligned} \det A &= 1\cdot\det\begin{pmatrix}1&0\\4&1\end{pmatrix} +2\cdot\det\begin{pmatrix}3&1\\0&4\end{pmatrix}\\ &= 1(1)+2(12)=25. \end{aligned}\begin{aligned} \det A &= 1\cdot\det\begin{pmatrix}1&0\\4&1\end{pmatrix} +2\cdot\det\begin{pmatrix}3&1\\0&4\end{pmatrix}\\ &= 1(1)+2(12)=25. \end{aligned}

Thus $\det A \neq 0$ , so $A$ is invertible.

7Relation to rank and linear maps

For an $n \times n$ square matrix,

A isinvertible ⟺ \det A \neq 0 ⟺ rank (A) = n ⟺ thecolumnsarelinearlyindependent .

The determinant proof uses the adjugate formula above. The rank viewpoint says that full rank means the column space is all of $K^{n}$ , so the corresponding linear map loses no information.

data/lecture/math/linear-algebra/rank-basics.lecture.n.md data/lecture/math/linear-algebra/rank-and-nullity-of-linear-maps.lecture.n.md

8How to choose the method

Use cofactor expansion when a row or column has many zeros. Use the adjugate when proving the inverse formula. For large numerical computations, row operations are usually more efficient than cofactor expansion.

9Scope

This lecture treats square matrices over the real or complex numbers. Cofactor expansion applies to every square matrix, but it becomes computationally expensive for large matrices. Rectangular matrices do not have this determinant or cofactor expansion in the same sense; rank and one-sided inverses are used instead.

10Final forms

[PARSE ERROR: Undefined("Command(\"boxed\")")] C_{i j} = (- 1)^{i + j} \det A^{(i | j)}

[PARSE ERROR: Undefined("Command(\"boxed\")")] \det A = \sum_{j = 1}^{n} a_{i j} C_{i j}

[PARSE ERROR: Undefined("Command(\"boxed\")")] A adj (A) = adj (A) A = (\det A) I

[PARSE ERROR: Undefined("Command(\"boxed\")")] A^{- 1} = \frac{1}{\det A} adj (A) (\det A \neq 0)

11In one sentence

Cofactor expansion groups the permutation definition by one chosen row or column.
The adjugate constructs the inverse when the determinant is nonzero.

12Exercise links

data/exercise/math/linear-algebra/determinants-and-invertibility.exercise.n.md