Adjugate
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In linear algebra, the adjugate of a square matrix is a matrix which plays a role similar to the inverse of a matrix; it can however be defined for any square matrix without the need to perform any divisions.
The adjugate has sometimes been called the "adjoint", but that terminology is ambiguous and is not used in Wikipedia. Today, "adjoint" normally refers to the conjugate transpose.
Definition
Suppose R is a commutative ring and A is an n-by-n matrix with entries from R. The adjugate of A, written as adj(A), is the n-by-n matrix whose entry in row j and column i is given by
- (-1)i+j Mij = Cij
where Mij represents the (n-1)×(n-1) minor of A (obtained by deleting row i and column j of A and taking the determinant of the resulting (n-1)-by(n-1) matrix), and Cij represents the matrix cofactors.
Example
As an example, we have
- <math>\operatorname{adj}\begin{pmatrix}
2& 1&0\\ 1&-1&1\\ 0&2&-1 \end{pmatrix}= \begin{pmatrix} -1&1&1\\ 1&-2&-2\\ 2&-4&-3 \end{pmatrix}. <math>
Here, the -4 in the last row, second column was computed by deleting the second row and last column of the original matrix and computing
- <math>(-1)^{3+2}\;\operatorname{det}\begin{pmatrix}2&1\\
0&2 \end{pmatrix}=(-1)(4)=-4. <math>
Properties
As a consequence of Laplace's formula for the computation of determinants, we have
- A adj(A) = adj(A) A = det(A) In
where In denotes the n-by-n identity matrix. This formula is used to prove that A is invertible as a matrix over R if and only if det(A) is invertible as an element of R. As another consequence of this, we can find the inverse easily from the adjugate - multiplying adjugate by the inverse of the determinant yields the identity matrix. From the above equation this is clear by multiplying throughout by A-1/det(A):
- A-1 = adj(A) / det(A).
We have the properties
- adj(In) = In
and
- adj(AB) = adj(B) adj(A)
for all n-by-n matrices A and B. The adjugate is also compatible with transposition:
- adj(AT) = (adj(A))T.
Furthermore,
- det(adj(A)) = det(A)n-1.
If p(t) = det(A - tIn) is the characteristic polynomial of A and we define the polynomial q(t) = (p(0) - p(t))/t, then
- adj(A) = q(A).
The adjugate also appears in the formula of the derivative of the determinant.