In linear algebra, the adjugate or classical adjoint of a square matrix is the transpose of its cofactor matrix and is denoted by . It is also occasionally known as adjunct matrix, or "adjoint", though the latter term today normally refers to a different concept, the adjoint operator which for a matrix is the conjugate transpose.
The product of a matrix with its adjugate gives a diagonal matrix (entries not on the main diagonal are zero) whose diagonal entries are the determinant of the original matrix:
where is the identity matrix of the same size as . Consequently, the multiplicative inverse of an invertible matrix can be found by dividing its adjugate by its determinant.
Definition
The adjugate of is the transpose of the cofactor matrix of ,
In more detail, suppose is a unital commutative ring and is an matrix with entries from . The -minor of , denoted , is the determinant of the matrix that results from deleting row and column of . The cofactor matrix of is the matrix whose entry is the cofactor of , which is the -minor times a sign factor:
The adjugate of is the transpose of , that is, the matrix whose entry is the cofactor of ,
Important consequence
The adjugate is defined so that the product of with its adjugate yields a diagonal matrix whose diagonal entries are the determinant . That is,
where is the identity matrix. This is a consequence of the Laplace expansion of the determinant.
The above formula implies one of the fundamental results in matrix algebra, that is invertible if and only if is an invertible element of . When this holds, the equation above yields
Examples
1 × 1 generic matrix
Since the determinant of a 0 × 0 matrix is 1, the adjugate of any 1 × 1 matrix (complex scalar) is . Observe that
2 × 2 generic matrix
The adjugate of the 2 × 2 matrix
is
By direct computation,
In this case, it is also true that ((A)) = (A) and hence that ((A)) = A.
3 × 3 generic matrix
Consider a 3 × 3 matrix
Its cofactor matrix is
where
Its adjugate is the transpose of its cofactor matrix,
3 × 3 numeric matrix
As a specific example, we have
It is easy to check the adjugate is the inverse times the determinant, .
The in the second row, third column of the adjugate was computed as follows. The (2,3) entry of the adjugate is the (3,2) cofactor of A. This cofactor is computed using the submatrix obtained by deleting the third row and second column of the original matrix A,
The (3,2) cofactor is a sign times the determinant of this submatrix:
and this is the (2,3) entry of the adjugate.
Properties
For any matrix , elementary computations show that adjugates have the following properties: , where is the identity matrix. , where is the zero matrix, except that if then . for any scalar . . . If is invertible, then . It follows that: is invertible with inverse . . is entrywise polynomial in . In particular, over the real or complex numbers, the adjugate is a smooth function of the entries of .
Over the complex numbers, , where the bar denotes complex conjugation. , where the asterisk denotes conjugate transpose.
Suppose that is another matrix. Then
This can be proved in three ways. One way, valid for any commutative ring, is a direct computation using the Cauchy–Binet formula. The second way, valid for the real or complex numbers, is to first observe that for invertible matrices and ,
Because every non-invertible matrix is the limit of invertible matrices, continuity of the adjugate then implies that the formula remains true when one of or is not invertible.
A corollary of the previous formula is that, for any non-negative integer ,
If is invertible, then the above formula also holds for negative .
From the identity
we deduce
Suppose that commutes with . Multiplying the identity on the left and right by proves that
If is invertible, this implies that also commutes with . Over the real or complex numbers, continuity implies that commutes with even when is not invertible.