Chapter 14. Cofactors and Adjugates

Cofactors and adjugates give a structured way to express determinants, inverses, and solutions of square systems. They are built from minors. A minor is a smaller determinant obtained by deleting one row and one column. A cofactor is a signed minor. The adjugate is the transpose of the cofactor matrix. For a square matrix $A$, the central identity is

When $\det(A)\ne 0$, this identity gives

This formula is important theoretically. It explains why determinants control invertibility and why entries of an inverse are ratios of determinants. The adjugate is the transpose of the cofactor matrix, and it satisfies the determinant identity above. (en.wikipedia.org)

14.1 Minors

Let $A$ be an $n\times n$ matrix. The minor $M_{ij}$ is the determinant of the $(n-1)\times(n-1)$ matrix obtained by deleting row $i$ and column $j$ from $A$.

For example, let

To compute $M_{12}$, delete row $1$ and column $2$. The remaining matrix is

Thus

A minor is always a determinant. It is not the smaller matrix itself.

14.2 Cofactors

The cofactor $C_{ij}$ is the signed minor

The sign factor alternates according to the checkerboard pattern

For the previous matrix,

Since

we have

The cofactor differs from the minor only by this sign.

14.3 Cofactor Matrix

The cofactor matrix of $A$ is the matrix whose $(i,j)$-entry is $C_{ij}$:

The cofactor matrix has the same size as $A$. Each entry records the determinant of the complementary submatrix, with the correct sign.

For a $2\times 2$ matrix

the cofactor matrix is

Indeed,

14.4 The Adjugate Matrix

The adjugate of $A$, written

is the transpose of the cofactor matrix:

Thus the $(i,j)$-entry of $\operatorname{adj}(A)$ is

For

the cofactor matrix is

so the adjugate is

This is the familiar matrix appearing in the $2\times 2$ inverse formula.

14.5 Cofactor Expansion

The determinant of $A$ may be expanded along any row or column.

Expansion along row $i$ is

Expansion along column $j$ is

This is called cofactor expansion or Laplace expansion. It expresses a determinant in terms of smaller determinants. (en.wikipedia.org)

For example, if

expanding along the first row gives

Compute

and

Therefore

14.6 Expansion Along a Different Row or Column

Cofactor expansion gives the same determinant no matter which row or column is chosen.

A useful strategy is to expand along a row or column with many zeros.

For example,

Expand along row $2$:

Now

Thus

Therefore

The zeros remove two terms from the expansion.

14.7 The Adjugate Identity

The fundamental identity for the adjugate is

Also,

These identities hold for every square matrix $A$, whether or not $A$ is invertible. (en.wikipedia.org)

The diagonal entries of $A\operatorname{adj}(A)$ are cofactor expansions of $\det(A)$. The off-diagonal entries are cofactor expansions of matrices with two equal rows, whose determinants are zero.

This explains why the product is a scalar multiple of the identity.

14.8 Proof Idea for the Adjugate Identity

Let $A=(a_{ij})$, and let $C=(C_{ij})$ be its cofactor matrix. Since

the $(i,k)$-entry of

is

If $i=k$, this is the cofactor expansion of $\det(A)$ along row $i$:

If $i\ne k$, the sum is the determinant of a matrix obtained from $A$ by replacing row $k$ with row $i$. That matrix has two equal rows, so its determinant is zero.

Thus

has $\det(A)$ on the diagonal and $0$ off the diagonal. Hence

14.9 Inverse Formula

If $A$ is invertible, then

From the adjugate identity,

Divide by $\det(A)$:

Therefore

This formula is valid exactly when $\det(A)\ne 0$. It gives a closed form for the inverse using determinants. (en.wikipedia.org)

14.10 Example: $2\times 2$ Inverse

Let

Then

Since

the inverse is

provided

For example,

Then

Thus

14.11 Example: Cofactor Matrix of a $3\times 3$ Matrix

Let

Compute the cofactors.

First row:

Second row:

Third row:

Thus the cofactor matrix is

14.12 Example: Adjugate and Inverse

Using the same matrix

the adjugate is the transpose of the cofactor matrix:

The determinant may be computed from the first row:

Thus

Therefore

This agrees with the general formula

14.13 Cofactors and Cramer’s Rule

Cofactors also explain Cramer’s rule.

For an invertible system

the solution is

Using the adjugate formula,

The $i$-th component of this vector can be written as

where $A_i(b)$ is obtained by replacing column $i$ of $A$ by $b$.

This gives Cramer’s rule. It is mainly a theoretical formula, since elimination is more efficient for large systems.

14.14 Adjugate of a Product

If $A$ and $B$ are square matrices of the same size, then

The order reverses, just as it does for inverses:

When $A$ and $B$ are invertible, the formula follows from

The identity also holds in general. (en.wikipedia.org)

14.15 Transpose and Adjugate

The adjugate interacts naturally with transpose:

This follows from the fact that minors of $A^T$ correspond to transposed minors of $A$, and determinants are unchanged by transpose.

For real matrices, this formula is often enough. For complex matrices, one must distinguish transpose from conjugate transpose. The classical adjugate uses cofactors and transpose, not the Hilbert-space adjoint.

14.16 Scalar Multiples

If $A$ is $n\times n$ and $c$ is a scalar, then

The exponent is $n-1$ because each cofactor is the determinant of an $(n-1)\times(n-1)$ matrix. Multiplying $A$ by $c$ multiplies each such submatrix by $c$, so each cofactor is multiplied by

14.17 Determinant of the Adjugate

If $A$ is $n\times n$, then

When $A$ is invertible, this follows from

Taking determinants gives

Since scaling an $n\times n$ matrix by $\det(A)$ multiplies its determinant by $\det(A)^n$,

14.18 Singular Matrices

The adjugate is defined even when $A$ is singular.

If

then the adjugate identity becomes

This does not mean that $\operatorname{adj}(A)$ must be the zero matrix. It means that every column of $\operatorname{adj}(A)$ lies in the null space of $A$.

For example,

Then

The determinant is zero, and indeed

Since $A$ is singular, the inverse formula cannot be used.

14.19 Rank and the Adjugate

The adjugate reflects the rank of a square matrix.

For an $n\times n$ matrix $A$:

This follows from the fact that entries of the adjugate are $(n-1)\times(n-1)$ minors. If all such minors vanish, the adjugate is zero.

14.20 Computational Role

Cofactors and adjugates are valuable for theory, symbolic formulas, and small matrices.

For large numerical matrices, they are usually not the preferred method for computing inverses. Cofactor expansion grows rapidly in cost. Row reduction, LU decomposition, QR decomposition, and other factorizations are more efficient and numerically better behaved.

Thus the adjugate formula should be understood as a structural identity first and a computational method only for small or symbolic cases.

14.21 Common Mistakes

14.22 Summary

Minors, cofactors, and adjugates refine the determinant into matrix form.

The main definitions are:

The central identity is

When $\det(A)\ne 0$, it gives

Cofactors and adjugates provide the algebraic bridge from determinants to inverses, Cramer’s rule, and symbolic matrix formulas.