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

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

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:

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

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

Cofactors and the Sign Pattern

The cofactor

C_{ij}

attaches a prescribed sign to the minor:

Cofactor Definition

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

Cofactor definition →

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:

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

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.

Using the

3 \times 3

matrix from the previous section, the cofactors are:

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

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

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

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.

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

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

:

\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:

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

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

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

\det(A) = 1(23) + 3(-12) + 0 + 2(-17) = 23 - 36 - 34 = -47

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

-47

, 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

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

Column

2

has two zeros. Expanding along column

2

:

\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

:

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.

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,

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

the cofactor matrix is

\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.

The cofactor matrix encodes every possible cofactor expansion simultaneously — each row of

\text{cof}(A)

contains the cofactors needed for expansion along the corresponding row of

A

, and each column contains those needed for column expansion.

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:

\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 →

To see why, consider the

(i,k)

entry of the product

A \cdot \operatorname{adj}(A)

. This is

\sum_{j=1}^{n} a_{ij} \cdot [\operatorname{adj}(A)]_{jk} = \sum_{j=1}^{n} a_{ij} \, C_{kj}

. When

i = k

, this sum is exactly the Laplace expansion of

\det(A)

along row

i

, so the diagonal entries equal

\det(A)

. When

i \neq k

, the sum pairs the entries of row

i

with the cofactors of a different row

k

. This is equivalent to computing the determinant of a matrix with two identical rows (row

i

appears in both its own position and row

k

's), which is always zero. So the off-diagonal entries vanish.

Verification

With

\det(A) = -12

:

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

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

.

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.

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 →

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

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?

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

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?

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

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