# Adjugate matrix

In linear algebra, the **adjugate**, **classical adjoint**, or **adjunct** of a square matrix is the *transpose of its cofactor matrix*.^{[1]}

The adjugate^{[2]} has sometimes been called the "adjoint",^{[3]} but today the "adjoint" of a matrix normally refers to its corresponding adjoint operator, which is its conjugate transpose.

## Definition

The adjugate of **A** is the *transpose of the cofactor matrix* **C** of **A**,

In more detail, suppose *R* is a commutative ring and **A** is an *n*×*n* matrix with entries from *R*.

- The (
*i*,*j*)*minor*of**A**, denoted**M**_{ij}, is the determinant of the (*n*− 1)×(*n*− 1) matrix that results from deleting row i and column j of**A**. - The cofactor matrix of
**A**is the*n*×*n*matrix**C**whose (*i*,*j*) entry is the (*i*,*j*)*cofactor*of**A**,

- The adjugate of
**A**is the*transpose*of**C**, that is, the*n*×*n*matrix whose (*i*,*j*) entry is the (*j*,*i*) cofactor of**A**,

- .

The adjugate is defined as it is so that the product of **A** with its adjugate yields a diagonal matrix whose diagonal entries are det(**A**),

**A** is invertible if and only if det(**A**) is an invertible element of *R*, and in that case the equation above yields

## Examples

### 1 × 1 generic matrix

The adjugate of any 1×1 matrix is .

### 2 × 2 generic matrix

The adjugate of the 2×2 matrix

is
.
It is seen that det(adj(**A**)) = det(**A**) and hence that adj(adj(**A**)) = **A**.

### 3 × 3 generic matrix

Consider a 3×3 matrix

is

where

- .

Its adjugate is the transpose of its cofactor matrix:

### 3 × 3 numeric matrix

As a specific example, we have

- .

The −6 in the third row, second column of the adjugate was computed as follows:

Again, the (3,2) entry of the adjugate is the (2,3) cofactor of *A*. Thus, the submatrix

was obtained by deleting the second row and third column of the original matrix **A**.

It is easy to check the adjugate is the inverse times the determinant, −6.

## Properties

The adjugate has the properties

for *n*×*n* matrices **A** and **B**. The second line follows from equations adj(**B**)adj(**A**) =
det(**B**)**B**^{−1} det(**A**)**A**^{−1} = det(**AB**)(**AB**)^{−1}.

Substituting in the second line **B** = **A**^{m − 1} and performing the recursion, one finds, for all integer *m*,

The adjugate preserves transposition,

Furthermore,

- If
**A**is a*n*×*n*matrix with*n*≥ 2, then and - If
**A**is an invertible*n*×*n*matrix, then

so that, if *n = 2* and **A** is invertible, then det(adj(**A**)) = det(**A**) and adj(adj(**A**)) = **A**.

Taking the adjugate of an invertible matrix **A** k times yields

### Inverses

In consequence of Laplace's formula for the determinant of an *n*×*n* matrix **A**,

where is the *n*×*n* identity matrix. Indeed, the (*i*,*i*) entry of the product **A** adj(**A**) is the scalar product of row *i* of **A** with row *i* of the cofactor matrix **C**, which is simply the Laplace formula for det(**A**) expanded by row *i*.

Moreover, for *i* ≠ *j* the (*i*,*j*) entry of the product is the scalar product of row *i* of **A** with row *j* of **C**, which is the Laplace formula for the determinant of a matrix whose *i* and *j* rows are equal, and therefore vanishes.

From this formula follows one of the central results in matrix algebra: A matrix **A** over a commutative ring R is invertible if and only if det(**A**) is invertible in R.

For, if **A** is an invertible matrix, then

and equation (*) above implies

Similarly, the **resolvent** of **A** is

where *p*(*t*) is the characteristic polynomial of **A**.

### Characteristic polynomial

If

is the characteristic polynomial of the matrix n-by-n matrix with coefficients in the ring R, then

where

is the first divided difference of p, a symmetric polynomial of degree n−1.

### Jacobi's formula

The adjugate also appears in Jacobi's formula for the derivative of the determinant,

### Cayley–Hamilton formula

The Cayley–Hamilton theorem allows the adjugate of **A** to be represented in terms of traces and powers of **A**:

where *n* is the dimension of **A**, and the sum is taken over *s* and all sequences of *k _{l}* ≥ 0 satisfying the linear Diophantine equation

For the 2×2 case this gives

For the 3×3 case this gives

For the 4×4 case this gives

The same results follow directly from the terminating step of the fast Faddeev–LeVerrier algorithm.

## See also

## References

- ↑ Gantmacher, F. R. (1960).
*The Theory of Matrices*.**1**. New York: Chelsea. pp. 76–89. ISBN 0-8218-1376-5. - ↑ Strang, Gilbert (1988). "Section 4.4: Applications of determinants".
*Linear Algebra and its Applications*(3rd ed.). Harcourt Brace Jovanovich. pp. 231–232. ISBN 0-15-551005-3. - ↑ Householder, Alston S. (2006).
*The Theory of Matrices in Numerical Analysis*. Dover Books on Mathematics. pp. 166–168. ISBN 0-486-44972-6.

- Roger A. Horn and Charles R. Johnson (1991),
*Topics in Matrix Analysis*. Cambridge University Press, ISBN 978-0-521-46713-1

## External links

- Matrix Reference Manual
- Online matrix calculator (determinant, track, inverse, adjoint, transpose) Compute Adjugate matrix up to order 8
- "adjugate of { { a, b, c }, { d, e, f }, { g, h, i } }".
*Wolfram Alpha*.