Cofactors: Minors, Laplace Expansion & Adjugate

Minors

Given an

n \times n

matrix

A

, the

(i,j)

minor

M_{ij}

is the determinant of the

(n-1) \times (n-1)

submatrix that remains after removing row

i

and column

j

. The minor is itself a determinant — a number, not a matrix.

Formally:

Minor Definition

M_{ij} = \det\!\left(A^{(i,j)}\right)

Minor definition →

where

A^{(i,j)}

denotes the submatrix of

A

with row

i

and column

j

removed.

For a

3 \times 3

matrix

Minors

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

Fact: Lewis Carroll's four-minor shortcut

In 1866 Charles Dodgson, the author of Alice's Adventures in Wonderland, published a way to build a determinant from smaller minors alone. For this matrix it reads

a_{22} \det(A) = M_{11} M_{33} - M_{13} M_{31}

: four corner minors give

20 \cdot 6 - (-12)(-13) = -36

, and dividing by the centre entry

3

gives

\det(A) = -12

.

there are nine minors, one for each entry. Deleting row

1

and column

1

leaves

\begin{pmatrix} 3 & -2 \\ 1 & 6 \end{pmatrix}

, so

M_{11} = 3 \cdot 6 - (-2) \cdot 1 = 20

. Deleting row

1

and column

2

leaves

\begin{pmatrix} 0 & -2 \\ 4 & 6 \end{pmatrix}

, so

M_{12} = 0 \cdot 6 - (-2) \cdot 4 = 8

. Deleting row

1

and column

3

leaves

\begin{pmatrix} 0 & 3 \\ 4 & 1 \end{pmatrix}

, so

M_{13} = 0 \cdot 1 - 3 \cdot 4 = -12

.

Continuing this way produces all nine values:

Minors

M_{21} = 5 \cdot 6 - 1 \cdot 1 = 29, \quad M_{22} = 2 \cdot 6 - 1 \cdot 4 = 8, \quad M_{23} = 2 \cdot 1 - 5 \cdot 4 = -18

Minors

M_{31} = 5 \cdot (-2) - 1 \cdot 3 = -13, \quad M_{32} = 2 \cdot (-2) - 1 \cdot 0 = -4, \quad M_{33} = 2 \cdot 3 - 5 \cdot 0 = 6

For a

4 \times 4

matrix, each minor is a

3 \times 3

determinant. For a

5 \times 5

matrix, each minor is

4 \times 4

. The recursive chain continues until reaching

1 \times 1

sub-determinants, where the minor is simply the lone entry.

Cofactor Notation

Notation

Cofactor Notation

Capital letters that secretly name numbers, a superscript that deletes instead of multiplying, and an operator with a dangerous old name.

\det

and its bars — determinant notation;

(a_{ij})

and

A^{T}

— matrix notation;

\Sigma

— sequence notation.

M_{ij}

·

C_{ij}

The minor and cofactor at position i, j

A convention wrinkle: capitals usually name matrices, but

M_{ij}

and

C_{ij}

are numbers — determinants of submatrices, defined in Minors and Cofactors and the Sign Pattern above and below. The double subscript locates which entry they belong to; the capital honours their determinant ancestry.

CasesThe letters regroup into genuine matrices one level up: the cofactor matrix

C = (C_{ij})

collects all

n^2

cofactors — a matrix whose entries are the capital-named scalars.

Also written

A_{ij}

for the cofactor in many Russian and older texts — colliding head-on with the entry notation

a_{ij}

, which is why

C_{ij}

won.

Do not confuseSubmatrices.

M_{ij}

is not the deleted-row-and-column matrix — it is that matrix's determinant; the matrix itself needs its own mark, next entry.

A^{(i,j)}

A with row i and column j removed

A parenthesized superscript that deletes: the

(n-1) \times (n-1)

submatrix left after striking row

i

and column

j

. The parentheses are load-bearing — they fence the indices off from the exponent reading.

CasesThe same fencing device appears wherever a superscript must not mean a power: the

n

-th derivative

f^{(n)}

uses identical armor. Un-parenthesized,

A^{ij}

would collide with powers and tensor conventions at once.

Also written

A_{\hat{i}\hat{j}}

with deletion hats, or verbal “the

(i,j)

submatrix” — no universal standard exists; every text declares its own mark.

Do not confuseA power or an entry. Three decorations, three meanings:

a_{ij}

selects an entry,

A^n

multiplies,

A^{(i,j)}

deletes — subscript, superscript, and fenced superscript all working different jobs on one letter.

\operatorname{adj}(A) = C^{T}

The adjugate of A — the transpose of the cofactor matrix

The transpose inside the definition is the notation trap: the adjugate is not the cofactor matrix but its transpose, and forgetting the flip is the classic error in the inverse formula of The Adjugate below.

CasesIts one starring role:

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

— the closed-form inverse, where the adjugate carries all the cofactor bookkeeping.

Also written

\operatorname{adj} A

without parentheses; older texts say “classical adjoint” and write

\operatorname{adj}

the same way.

Do not confuseThe modern adjoint. In advanced linear algebra “adjoint” means the conjugate transpose

A^{*}

— an unrelated operation. The name migrated; the word “adjugate” exists precisely to escape the collision, and careful reading of older texts must check which one “adjoint” means.

All linear algebra symbols →↑ Determinants

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Notation · Cheat sheet

Determinant Notation at a Glance

The marks from the notation cards across the determinant pages: how each is read and what it means. Every group links to the full card on its page.

Properties

\det(A)

the determinant of A

A number from a square matrix; brackets optional, as in

\det A

.

|A|

determinant of A, with bars

Bars around the name, or bars replacing the brackets of the array.

\Delta

delta

The school name for the determinant of a system.

CofactorsThis page

M_{ij}, \; C_{ij}

minor and cofactor at i, j

Numbers, despite the capitals:

C_{ij} = (-1)^{i+j}M_{ij}

.

A^{(i,j)}

A with row i and column j removed

The smaller matrix whose determinant is

M_{ij}

.

\operatorname{adj}(A) = C^{T}

the adjugate of A

The transpose of the cofactor matrix, not the cofactor matrix itself.

Geometry

\det(A) < 0

a negative determinant

The map flips orientation.

|\det(A)|

absolute value of the determinant

The factor by which areas or volumes scale.

\det J_T

the Jacobian determinant

The local scale factor of a non-linear map.

Applications

x_i = \frac{\det(A_i)}{\det(A)}

Cramer's rule

A_i

is

A

with column

i

replaced by

\mathbf{b}

.

W(f_1, \dots, f_n)(x)

the Wronskian

A determinant of functions and their derivatives, at

x

.

\prod_{i < j} (x_j - x_i)

the Vandermonde product

One factor for every pair with

i

below

j

.

D_n

D sub n

Determinant of the

n \times n

matrix in a recurrence.

Same marks, different job

|A|

vs

|x|

Bars around a matrix give its determinant, which can be negative; around a number, its absolute value.

||A||

Double bars are a matrix norm, not the absolute value of

\det A

; that is

|\det(A)|

.

A_i

In Cramer’s rule a whole matrix, not a row, column or entry.

Matrix operation symbols table Determinants

Determinant Notation at a Glance·/linear-algebra/determinants/propertiesLearn Math Class

Cofactors and the Sign Pattern

With the minors in hand, the natural guess is that a determinant is each entry times its minor, added up.

Across row 1 of the matrix from the previous section, that means

2 \cdot M_{11} + 5 \cdot M_{12} + 1 \cdot M_{13}

— every entry weighted by the determinant of what is left when its row and column are removed.

The plain sum comes out to 68, but the determinant is −12.

row-1 entry	its minor	entry × minor	with the checkerboard sign
a₁₁ = 2	M₁₁ = 20	40	+40
a₁₂ = 5	M₁₂ = 8	40	−40
a₁₃ = 1	M₁₃ = −12	−12	−12
total		68 — wrong	−12 — the determinant

Flipping the sign of the middle term turns

68

into

-12

, the correct value.

The failure was predictable: even the 2 × 2 determinant ad − bc already carries a minus sign.

Expanding

\begin{pmatrix} a & b \\ c & d \end{pmatrix}

along its first row gives

a \cdot d

and

b \cdot c

, and they must be subtracted, not added. Larger determinants inherit that alternation, like the colours of a chessboard: neighbouring positions always have opposite signs.

The exponent

i + j

determines whether the sign is positive or negative. When

i + j

is even, the cofactor equals the minor; when

i + j

is odd, the cofactor is the negative of the minor. This produces a checkerboard of signs across the matrix:

Cofactors and the Sign Pattern

\begin{pmatrix} + & - & + & - \\ - & + & - & + \\ + & - & + & - \\ - & + & - & + \end{pmatrix}

The definition

The cofactor

C_{ij}

is the minor

M_{ij}

with the sign of position

(i, j)

on the checkerboard attached:

C_{ij} = (-1)^{i+j} M_{ij}

. Entries times their cofactors add up to the determinant.

The cofactor

C_{ij}

attaches a prescribed sign to the minor:

Cofactor Definition

C_{ij} = (-1)^{i+j} \, M_{ij}

Cofactor definition →

So

C_{12} = (-1)^{3} \cdot 8 = -8

, and

2 \cdot 20 + 5 \cdot (-8) + 1 \cdot (-12) = -12

.

A cofactor is a signed minor: one number for each position of the matrix.

Using the

3 \times 3

matrix from the previous section, the cofactors are:

Cofactors and the Sign Pattern

C_{11} = (+1)(20) = 20, \quad C_{12} = (-1)(8) = -8, \quad C_{13} = (+1)(-12) = -12

Cofactors and the Sign Pattern

C_{21} = (-1)(29) = -29, \quad C_{22} = (+1)(8) = 8, \quad C_{23} = (-1)(-18) = 18

Cofactors and the Sign Pattern

C_{31} = (+1)(-13) = -13, \quad C_{32} = (-1)(-4) = 4, \quad C_{33} = (+1)(6) = 6

Comparing cofactors to minors, entries at even-sum positions are unchanged while entries at odd-sum positions flip sign.

Position (i, j)	Minor M_ij	Sign (−1)^i+j	Cofactor C_ij
(1, 1)	20	+	20
(1, 2)	8	−	−8
(1, 3)	−12	+	−12
(2, 1)	29	−	−29
(2, 2)	8	+	8
(2, 3)	−18	−	18
(3, 1)	−13	+	−13
(3, 2)	−4	−	4
(3, 3)	6	+	6

Read backwards, the sign rule is a test: a sign depends only on position, never on the entries.

The pattern always starts with

+

at position

(1,1)

and alternates from there. The sign depends entirely on the position — the actual entries of the matrix play no role in determining it.

The alternating sign board

Signs alternate from the top-left corner outward, so a position's sign depends only on whether its row and column indices sum to an even or an odd number. The board is fixed, independent of the matrix's contents, and it is the only thing separating a minor from a cofactor. Generate it at other sizes on the determinant visualizer.

Because the board never changes, the sign of a cofactor can always be read off its position without any computation.

The full expansion of a determinant by cofactors is the subject of the next section, Laplace expansion along a row.

The rest of the page covers Laplace expansion along a row and along a column, the cofactor matrix, the adjugate, and the computational cost.

Laplace Expansion Along a Row

The determinant of

A

can be computed by selecting any row

i

and summing the products of each entry in that row with its cofactor:

Laplace Row Expansion

\det(A) = \sum_{j=1}^{n} a_{ij} \, C_{ij} \qquad \text{for any fixed row } i

Laplace row expansion →

Expanded with the explicit sign factor:

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

.

The remarkable fact is that every row produces the same number. Expanding along row

1

, row

2

, or row

n

all yield the same determinant. This is not obvious from the formula itself — the proof relies on the algebraic properties of the determinant or on the permutation-based definition.

Worked Example: 4×4 Matrix

Laplace Expansion Along a Row

A = \begin{pmatrix} 1 & 3 & 0 & 2 \\ -1 & 0 & 2 & 1 \\ 0 & 4 & -1 & 3 \\ 2 & 1 & 0 & -2 \end{pmatrix}

Expanding along row

1

:

Laplace Expansion Along a Row

\det(A) = 1 \cdot C_{11} + 3 \cdot C_{12} + 0 \cdot C_{13} + 2 \cdot C_{14}

The zero entry at position

(1,3)

eliminates one

3 \times 3

determinant entirely. The three remaining cofactors require expanding the sub-determinants:

Laplace Expansion Along a Row

M_{11} = \det\begin{pmatrix} 0 & 2 & 1 \\ 4 & -1 & 3 \\ 1 & 0 & -2 \end{pmatrix} = 0(2 - 0) - 2(-8 - 3) + 1(0 + 1) = 0 + 22 + 1 = 23

Laplace Expansion Along a Row

M_{12} = \det\begin{pmatrix} -1 & 2 & 1 \\ 0 & -1 & 3 \\ 2 & 0 & -2 \end{pmatrix} = -1(2 - 0) - 2(0 - 6) + 1(0 + 2) = -2 + 12 + 2 = 12

Laplace Expansion Along a Row

M_{14} = \det\begin{pmatrix} -1 & 0 & 2 \\ 0 & 4 & -1 \\ 2 & 1 & 0 \end{pmatrix} = -1(0 + 1) - 0(0 + 2) + 2(0 - 8) = -1 + 0 - 16 = -17

Applying the signs:

C_{11} = +23

,

C_{12} = -12

,

C_{14} = -(-17) = +17

. The determinant is

Laplace Expansion Along a Row

\det(A) = 1(23) + 3(-12) + 0 + 2(17) = 23 - 36 + 34 = 21

Pitfall: Dropping the sign factor

Using the minors in place of the cofactors gives

1(23) + 3(12) + 0 + 2(-17) = 25

. With the signs

+, -, +, -

of row

1

, the sum is

23 - 36 + 34 = 21

.

Verification via a Different Row

Expanding the same matrix along row

3

(which has a zero in the first position) would produce the same value

21

, confirming that the choice of row is purely a matter of computational convenience.

One row expanded, term by term

Each entry of the row is multiplied by the determinant of the matrix left when its own row and column are removed, and the sign board decides whether the term is added or subtracted. The recursion is visible: every minor is itself a determinant awaiting the same treatment. Expand along a column instead and check the answer matches on the determinant visualizer.

Expanding along a row with zeros in it kills those terms before their minors are ever computed, which is the only real economy available here.

Laplace Expansion Along a Column

The expansion formula works identically along columns. Fixing column

j

:

Laplace Column Expansion

\det(A) = \sum_{i=1}^{n} a_{ij} \, C_{ij} \qquad \text{for any fixed column } j

Laplace column expansion →

Expanded with the explicit sign factor:

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

.

That column expansion gives the same result as row expansion follows from transpose invariance: since

\det(A^T) = \det(A)

, expanding

A

along column

j

is the same as expanding

A^T

along row

j

.

The practical consequence is that before starting any cofactor expansion, the first step should be to scan the matrix for the row or column containing the most zeros. Each zero entry eliminates an entire sub-determinant from the sum.

Worked Example

Laplace Expansion Along a Column

B = \begin{pmatrix} 3 & 0 & 0 \\ 1 & -2 & 5 \\ 4 & 0 & 7 \end{pmatrix}

Column

2

has two zeros. Expanding along column

2

:

Laplace Expansion Along a Column

\det(B) = 0 \cdot C_{12} + (-2) \cdot C_{22} + 0 \cdot C_{32} = (-2) \cdot C_{22}

The minor

M_{22}

is the

2 \times 2

determinant from deleting row

2

and column

2

:

Laplace Expansion Along a Column

M_{22} = \det\begin{pmatrix} 3 & 0 \\ 4 & 7 \end{pmatrix} = 21

Since

C_{22} = (-1)^{2+2}(21) = 21

, we get

\det(B) = (-2)(21) = -42

.

An expansion along row

1

or column

1

would require more terms but produce the same result. The column

2

expansion reduced the work to a single

2 \times 2

determinant.

Expanding down column 1

The same expansion, turned ninety degrees. The pivot is now a₂₁, the second entry of the first column; its row and column are struck out, and what is left is the minor M₂₁, shown alongside. Its sign comes from the same checkerboard, (−1)²⁺¹ = −1. Summing a_i1C_i1 down the column gives exactly the determinant the row expansion gave. Run both directions on the determinant visualizer.

Whichever row or column is chosen, the answer is the same, so the practical choice is always the line with the most zeros.

The Cofactor Matrix

The cofactor matrix of

A

, sometimes written

\text{cof}(A)

, is the

n \times n

matrix whose

(i,j)

entry is the cofactor

C_{ij}

:

Cofactor Matrix Definition

\operatorname{cof}(A) = \bigl[C_{ij}\bigr]_{n \times n}

Cofactor matrix definition →

It is not the matrix of minors — the sign factors

(-1)^{i+j}

are already incorporated.

For the

3 \times 3

matrix used earlier,

The Cofactor Matrix

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

the cofactor matrix is

The Cofactor Matrix

\text{cof}(A) = \begin{pmatrix} 20 & -8 & -12 \\ -29 & 8 & 18 \\ -13 & 4 & 6 \end{pmatrix}

where each entry was computed in the earlier sections. As a check, the Laplace expansion along row

1

should give

\det(A) = 2(20) + 5(-8) + 1(-12) = 40 - 40 - 12 = -12

. Along row

2

:

0(-29) + 3(8) + (-2)(18) = 0 + 24 - 36 = -12

. Along row

3

:

4(-13) + 1(4) + 6(6) = -52 + 4 + 36 = -12

. All three rows agree.

Pitfall: Starting every row with a plus sign

Giving row

2

the signs

+, -, +

turns its expansion into

0(29) - 3(8) + (-2)(-18) = 12

. Row

2

starts with

(-1)^{2+1} = -1

, so its signs are

-, +, -

and the sum is

-12

.

The Adjugate

The adjugate (also called the classical adjoint) of

A

is the transpose of the cofactor matrix:

Adjugate Definition

\operatorname{adj}(A) = \operatorname{cof}(A)^T

Adjugate definition →

For the running example:

The Adjugate

\operatorname{adj}(A) = \begin{pmatrix} 20 & -29 & -13 \\ -8 & 8 & 4 \\ -12 & 18 & 6 \end{pmatrix}

The Fundamental Identity

The adjugate satisfies

Adjugate Identity

A \cdot \operatorname{adj}(A) = \operatorname{adj}(A) \cdot A = \det(A) \, I

Adjugate identity →

Pitfall: Forgetting to transpose the cofactors

Using

\operatorname{cof}(A)

itself as the adjugate pairs row

1

of

A

with column

1

of

\operatorname{cof}(A)

:

2(20) + 5(-29) + 1(-13) = -118

, not

-12

. After the transpose, row

1

meets its own cofactors:

2(20) + 5(-8) + 1(-12) = -12

.

Verification

With

\det(A) = -12

:

The Adjugate

A \cdot \operatorname{adj}(A) = \begin{pmatrix} 2 & 5 & 1 \\ 0 & 3 & -2 \\ 4 & 1 & 6 \end{pmatrix} \begin{pmatrix} 20 & -29 & -13 \\ -8 & 8 & 4 \\ -12 & 18 & 6 \end{pmatrix} = \begin{pmatrix} -12 & 0 & 0 \\ 0 & -12 & 0 \\ 0 & 0 & -12 \end{pmatrix} = -12 \, I

Fact: A singular adjugate has rank 1 or 0

When

\det(A) = 0

, the identity says

A \cdot \operatorname{adj}(A) = O

, so every column of

\operatorname{adj}(A)

lies in the null space of

A

. If that null space is a single line,

\operatorname{adj}(A)

has rank exactly

1

; if it is larger, every cofactor is

0

.

This identity is the foundation of the adjugate inverse formula: dividing both sides by

\det(A)

gives

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

, valid whenever

\det(A) \neq 0

.

The identity itself asks nothing of

\det(A)

. The adjugate is built entry by entry from the minors, so it exists for every square matrix, singular or not. For

\begin{pmatrix} 1 & 2 \\ 2 & 4 \end{pmatrix}

, whose determinant is

0

, the adjugate is

\begin{pmatrix} 4 & -2 \\ -2 & 1 \end{pmatrix}

, and the product of the two matrices is the zero matrix, exactly as

\det(A) \, I

predicts.

A row paired with its own cofactors gives the determinant; paired with another row's cofactors, it gives zero.

Computational Cost

Cofactor expansion is a recursive algorithm. Each

n \times n

determinant spawns

n

sub-problems of size

(n-1) \times (n-1)

. Without any zero entries to prune terms, the total number of multiplications satisfies the recurrence

T(n) = n \cdot T(n-1)

, which gives

T(n) = O(n!)

.

To put this in concrete terms: a

10 \times 10

determinant via cofactor expansion requires roughly

10! \approx 3.6

million multiplications. A

20 \times 20

determinant would require over

2 \times 10^{18}

— well beyond the reach of any computer running a naive recursive implementation. Row reduction, by contrast, computes the same determinant in roughly

\frac{2}{3}n^3

operations: about

670

for

n = 10

and about

5300

for

n = 20

.

This cost difference does not make cofactor expansion useless. For matrices up to

4 \times 4

, the expansion is fast enough to do by hand and gives the exact symbolic result. For matrices with many zero entries, the effective cost drops dramatically because each zero eliminates an entire recursive branch. In symbolic computation — where entries are polynomials or formal expressions rather than numbers — cofactor expansion preserves structure that row reduction would obscure.

The Laplace expansion is best understood as a theoretical instrument. It defines what the determinant is, establishes its algebraic properties, and produces the adjugate and the cofactor structure. For numerical computation on anything larger than a small matrix, the row-reduction approach is the practical choice.

Matrix size n	Cofactor expansion ≈ n!	Row reduction ≈ (2/3)n³	Ratio
4	24	≈ 43	cofactor faster
5	120	≈ 83	≈ 1.4×
10	≈ 3.6 million	≈ 670	≈ 5400×
20	≈ 2.4 × 10¹⁸	≈ 5,300	≈ 5 × 10¹⁴×

Summary: The Cofactor Construction Chain

The objects introduced above — minor, cofactor, cofactor matrix, adjugate — form a single construction chain, each built from the previous one and culminating in the inverse formula. The table below collects each link in the chain alongside its definition, its shape (number or matrix), and the role it plays in connecting one stage to the next.

These six are a chain rather than a list, and the order is not negotiable: a cofactor needs a minor, an expansion needs cofactors, the cofactor matrix collects them, the adjugate transposes it, and the inverse formula divides by the determinant. Each definition exists because the next one requires it, which is why the terminology is worth learning in one pass rather than six.

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Determinants · cofactors

The cofactor construction chain

Six definitions in strict order, each built from the one above it. Read alone they look like six pieces of terminology; read as a chain they are one construction, and the last step is what the first five were for.

6steps

Building a number3

1

Minor§ 1

a number; unsigned

M_{ij} = \det(A

with row

i

, column

j

deleted

)

The raw building block, and note it is itself a determinant — the definition is recursive, which is where the cost comes from. For a

3 \times 3

matrix each minor is a

2 \times 2

determinant; for a

4 \times 4

each is a

3 \times 3

, and so on down.

2

Cofactor§ 2

a number; signed

C_{ij} = (-1)^{i+j} M_{ij}

The minor with a checkerboard sign attached. The sign is not decoration — it is what makes the expansion below add up to the determinant rather than to nothing in particular, and dropping it is the most common error in the whole construction.

3

Laplace expansion§ 3

any single row or column

\det(A) = \sum_j a_{ij}C_{ij}

Where the cofactors do their work. Any row or column gives the same answer — a consequence of transpose invariance — so choose the one with the most zeros, since each zero entry removes an entire minor from the calculation.

Building a matrix3

4

Cofactor matrix§ 5

an

n \times n

matrix

\operatorname{cof}(A)_{ij} = C_{ij}

Every cofactor collected in position. On its own it does nothing; it exists to be transposed in the next step, which is the only reason to compute all

n^2

of them rather than the single row the expansion needs.

5

Adjugate§ 6

A \cdot \operatorname{adj}(A) = \det(A) \cdot I

\operatorname{adj}(A) = \operatorname{cof}(A)^{\mathsf{T}}

The transpose, and the identity beside it is the whole point of the chain. It holds for every square matrix including singular ones — when

\det(A) = 0

it says

A \cdot \operatorname{adj}(A) = O

, which is a real statement rather than a failure.

6

Inverse formula§ 6

\det(A) \neq 0

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

Divide the identity above by

\det(A)

and the inverse falls out. This is the end of the chain and its justification — an explicit formula for every entry of the inverse in terms of the entries of

A

.

Nothing in this chain is a good way to compute anything. A minor is a determinant, so each level multiplies the work by

n

— which is why row reduction computes determinants and factorization computes inverses. The chain earns its place by giving a closed form, not a method.

The cofactor construction chain·/linear-algebra/determinants/cofactorsLearn Math Class

The identity

A \cdot \operatorname{adj}(A) = \det(A) \cdot I

is the destination, and it is more general than the inverse formula it produces. It holds for every square matrix, singular ones included — where it says

A \cdot \operatorname{adj}(A) = O

, which is informative rather than vacuous. Dividing by

\det(A)

is the last step, and it is the only step that requires a hypothesis.

None of this is how anything is computed. A minor is a determinant, so every level of the recursion multiplies the work by another factor of

n

— the adjugate of a

5 \times 5

matrix means twenty-five

4 \times 4

determinants, each of which means four

3 \times 3

determinants. Reduction computes a determinant in

\tfrac{2}{3}n^3

operations and factorization inverts in

2n^3

. What the chain provides instead is a closed form, and closed forms are what proofs and symbolic work need.

Common Pitfalls

Three slips come up again and again in cofactor work. Each is marked with a box where it happens in the sections above; here each is worked through in full, with the wrong and the right working side by side.

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1Dropping the sign factor

The entries of the expansion row are multiplied by their minors

M_{ij}

instead of their cofactors

C_{ij}

. The checkerboard signs never enter, so every term at a position where

i + j

is odd comes out with the wrong sign.

Wrong turn

Error here

For the

4 \times 4

matrix of the worked example, expanding along row

1

with the minors

M_{11} = 23

,

M_{12} = 12

,

M_{14} = -17

,

1 \cdot 23 + 3 \cdot 12 + 0

+ 2 \cdot (-17) = 23 + 36 - 34

= 25

Right path

The positions

(1,1), (1,2), (1,3), (1,4)

carry the signs

+, -, +, -

, so

C_{12} = -12

and

C_{14} = +17

.

1 \cdot 23 + 3 \cdot (-12) + 0

+ 2 \cdot 17 = 23 - 36 + 34

= 21

Why it happensThe minors take all the effort, and once they are found the expansion looks finished. The

2 \times 2

minors already contain a minus sign of their own, in

ad - bc

, which makes it easy to believe the signs have been taken care of.

How to checkExpand along a second row. Along row

3

, which is

(0, 4, -1, 3)

, the minors are

M_{32} = -12

,

M_{33} = -3

,

M_{34} = 10

, and with the signs

-, +, -

at those positions the sum is

4 \cdot 12 + (-1)(-3) + 3 \cdot (-10) = 48 + 3 - 30 = 21

, matching row

1

. Without signs the same row gives

4(-12) + (-1)(-3) + 3(10) = -15

, which disagrees with the

25

from row

1

.

2Starting every row with a plus sign

The signs

+, -, +

are written down afresh at the start of every row or column instead of being read from the checkerboard. Along an odd-numbered row or column the pattern really does start with

+

, so the slip goes unnoticed there; along an even-numbered one every sign is reversed, and so is the determinant.

Start

For

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

,

Wrong turn

Error here

Row

2

has the minors

M_{21} = 29

,

M_{22} = 8

,

M_{23} = -18

. With the signs

+, -, +

,

0 \cdot 29 - 3 \cdot 8 + (-2)(-18)

= 0 - 24 + 36 = 12

Right path

The sign at position

(2,1)

is

(-1)^{2+1} = -1

, so row

2

runs

-, +, -

.

-0 \cdot 29 + 3 \cdot 8 - (-2)(-18)

= 0 + 24 - 36 = -12

Which matches the expansions along rows

1

and

3

.

Why it happensMost first examples are expanded along row

1

, where the pattern begins with

+

, and the habit is carried over to the other rows.

How to checkThe first sign of row

i

is

(-1)^{i+1}

, plus for odd rows and minus for even rows. If a second expansion gives the same size but the opposite sign, this slip is the usual cause.

3Forgetting to transpose the cofactors

The cofactor matrix itself is used as the adjugate. The identity

A \cdot \operatorname{adj}(A) = \det(A) \, I

then fails, and so does the inverse formula built on it. The slip is invisible only when

\operatorname{cof}(A)

happens to be symmetric.

Wrong turn

Error here

With the running example,

A \cdot \operatorname{cof}(A)

= \begin{pmatrix} 2 & 5 & 1 \\ 0 & 3 & -2 \\ 4 & 1 & 6 \end{pmatrix} \begin{pmatrix} 20 & -8 & -12 \\ -29 & 8 & 18 \\ -13 & 4 & 6 \end{pmatrix}

= \begin{pmatrix} -118 & 28 & 72 \\ -61 & 16 & 42 \\ -27 & 0 & 6 \end{pmatrix}

Which is not a multiple of

I

, so dividing it by

\det(A) = -12

does not give

A^{-1}

.

Right path

Transpose first,

\operatorname{adj}(A) = \operatorname{cof}(A)^T

. The product is then

-12 \, I

, as in the verification of the adjugate section.

Why it happensThe cofactor matrix is where all the work goes, and the transpose is a last step with no arithmetic in it. The cofactor

C_{ij}

belongs to position

(i,j)

of

A

, so it feels natural to leave it there.

How to checkEvery diagonal entry of

A

times the candidate adjugate is a Laplace expansion and must equal

\det(A)

. Row

2

of

A

against column

2

of the true adjugate gives

0(-29) + 3(8) + (-2)(18) = -12

; against column

2

of

\operatorname{cof}(A)

it gives

0(-8) + 3(8) + (-2)(4) = 16

.

Common pitfalls with cofactors·/linear-algebra/determinants/cofactorsLearn Math Class

Each slip loses a sign or a transpose. A cofactor is

(-1)^{i+j}

times its minor, the signs of each row are read from the checkerboard, and the adjugate is the transpose of the cofactor matrix. Expanding along a second row, or checking that

A \cdot \operatorname{adj}(A) = \det(A)\,I

, catches each slip.

Cofactors and Minors FAQ

Is the minor $M_{ij}$ a matrix or a number?

+

A number. Deleting row

i

and column

j

produces a smaller matrix, but

M_{ij}

names that matrix's determinant rather than the matrix itself. The submatrix needs its own notation if you want to refer to it. Treating

M_{ij}

as an array is a species error that surfaces as soon as you try to add or multiply with it.Read more →

Why is the superscript in $A^{(i,j)}$ written inside parentheses?

+

The parentheses are armour against misreading it as a power. Unfenced,

A^{ij}

would collide with exponents and with tensor index conventions at once. The same device appears in the

n

-th derivative

f^{(n)}

. Three decorations then do three jobs on one letter:

a_{ij}

selects an entry,

A^n

multiplies,

A^{(i,j)}

deletes.Read more →

Is the adjugate the same thing as the adjoint?

+

Not in modern usage. In advanced linear algebra the adjoint means the conjugate transpose

A^{*}

, an unrelated operation. The word adjugate exists precisely to escape that collision, since older texts did use adjoint for this cofactor construction. When reading an older source, check which meaning is intended before applying anything.Read more →

Does the adjugate exist when the determinant is zero?

+

Yes. Every entry of the adjugate is a signed minor, so it can be built for any square matrix, singular or not, and the identity

A \cdot \operatorname{adj}(A) = \det(A) \, I

still holds. With

\det(A) = 0

it reads

A \cdot \operatorname{adj}(A) = O

, so every column of the adjugate lies in the null space of

A

, and its rank drops to

1

or

0

. Only the step to the inverse fails, since it divides by

\det(A)

.Read more →

Why does a row times another row's cofactors give zero?

+

Because the sum is secretly the determinant of a matrix with two equal rows. The cofactors of row

k

never look at row

k

itself, so

\sum_j a_{ij} C_{kj}

is exactly the expansion along row

k

of the matrix in which row

k

has been overwritten by a copy of row

i

. Two identical rows force that determinant to

0

. This is what makes the off-diagonal entries of

A \cdot \operatorname{adj}(A)

vanish.Read more →

When is cofactor expansion better than row reduction?

+

When the matrix is small, sparse or symbolic. Up to about

4 \times 4

it is quick enough by hand and gives an exact result; each zero in the chosen row or column removes a whole branch of the recursion; and when the entries are polynomials or letters, it keeps the structure that elimination would bury in fractions. For a general numerical matrix it loses badly: about

10! \approx 3.6

million multiplications at

n = 10

, against roughly

670

operations for row reduction.Read more →

Minors, Cofactors, and the Adjugate

Expanding Along Any Row or Column

Minors

Cofactor Notation

Cofactor Notation

Cofactors and the Sign Pattern

Laplace Expansion Along a Row

Worked Example: 4×4 Matrix

Verification via a Different Row

Laplace Expansion Along a Column

Worked Example

The Cofactor Matrix

The Adjugate

The Fundamental Identity

Verification

Computational Cost

Summary: The Cofactor Construction Chain

The cofactor construction chain

Common Pitfalls

Cofactors and Minors FAQ

Is the minor $M_{ij}$ a matrix or a number?

Why is the superscript in $A^{(i,j)}$ written inside parentheses?

Is the adjugate the same thing as the adjoint?

Does the adjugate exist when the determinant is zero?

Why does a row times another row's cofactors give zero?

When is cofactor expansion better than row reduction?

Minors, Cofactors, and the Adjugate

Expanding Along Any Row or Column

Minors

Cofactor Notation

Cofactor Notation

Cofactors and the Sign Pattern

Laplace Expansion Along a Row

Worked Example: 4×4 Matrix

Verification via a Different Row

Laplace Expansion Along a Column

Worked Example

The Cofactor Matrix

The Adjugate

The Fundamental Identity

Verification

Computational Cost

Summary: The Cofactor Construction Chain

The cofactor construction chain

Common Pitfalls

Cofactors and Minors FAQ

Is the minor MijM_{ij}Mij​ a matrix or a number?

Why is the superscript in A(i,j)A^{(i,j)}A(i,j) written inside parentheses?

Is the adjugate the same thing as the adjoint?

Does the adjugate exist when the determinant is zero?

Why does a row times another row's cofactors give zero?

When is cofactor expansion better than row reduction?

Is the minor $M_{ij}$ a matrix or a number?

Why is the superscript in $A^{(i,j)}$ written inside parentheses?