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Concept
Adjugate matrix
Summary
In linear algebra, the adjugate or classical adjoint of a square matrix A is the transpose of its cofactor matrix and is denoted by adj(A). 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 I is the identity matrix of the same size as A. Consequently, the multiplicative inverse of an invertible matrix can be found by dividing its adjugate by its determinant.
The adjugate of A is the transpose of the cofactor matrix C of A,
In more detail, suppose R is a unital 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, which is the (i, j)-minor times a sign factor:
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 so that the product of A with its adjugate yields a diagonal matrix whose diagonal entries are the determinant det(A). That is,
where I is the n × n 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 A is invertible if and only if det(A) is an invertible element of R. When this holds, the equation above yields
Since the determinant of a 0 x 0 matrix is 1, the adjugate of any 1 × 1 matrix (complex scalar) is . Observe that
The adjugate of the 2 × 2 matrix
is
By direct computation,
In this case, it is also true that det(adj(A)) = det(A) and hence that adj(adj(A)) = A.
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