In linear algebra, the adjugate or classical adjoint of a square matrix is a matrix that 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. Today, "adjoint" of a matrix normally refers to its corresponding adjoint operator, which is its conjugate transpose.
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Suppose R is a commutative ring and A is an n×n matrix with entries from R. The definition of the adjugate of A is a multi-step process:

The adjugate of A is the transpose of the cofactor matrix of A:
.That is, the adjugate of A is the n×n matrix whose (i,j) entry is the (j,i) cofactor of A:
.The adjugate of the 2 × 2 matrix

is
.Consider the
matrix

Its adjugate is the transpose of the cofactor matrix

So that we have

where
.Therefore C, the adjugate of A, is

Note that the adjugate is the transpose of the cofactor matrix. Thus, for instance, the (3,2) entry of the adjugate is the (2,3) cofactor of A.
As a specific example, we have[1]
.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.
As a consequence of Laplace's formula for the determinant of an n×n matrix A, we have

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 is therefore zero.
From this formula follows one of the most important 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 if det(A) is a unit then (*) above shows that

See also Cramer's rule.
The adjugate has the properties


for all n×n matrices A and B.
The adjugate preserves transposition:
.Furthermore

If p(t) = det(A − t I) is the characteristic polynomial of A and we define the polynomial q(t) = (p(0) − p(t))/t, then

where
are the coefficients of p(t),

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

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