Adjugate matrix

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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[edit]

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 Mij, 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[edit]

1 × 1 generic matrix[edit]

The adjugate of any 1×1 matrix is .

2 × 2 generic matrix[edit]

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[edit]

Consider a 3×3 matrix

is

where

.

Its adjugate is the transpose of its cofactor matrix:

3 × 3 numeric matrix[edit]

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[edit]

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 = Am − 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[edit]

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 ij 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[edit]

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[edit]

Main article: Jacobi's formula

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

Cayley–Hamilton formula[edit]

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 kl ≥ 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[edit]

References[edit]

  1. ^ Gantmacher, F. R. (1960). The Theory of Matrices. 1. New York: Chelsea. pp. 76–89. ISBN 0-8218-1376-5. 
  2. ^ 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. 
  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[edit]