![]() 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:Ī adj ( A ) = det ( A ) I, īecause every non-invertible matrix is the limit of invertible matrices, continuity of the adjugate then implies that the formula remains true when one of A or B is not invertible.Ī corollary of the previous formula is that, for any non-negative integer k,Īdj ( A k ) = adj ( A ) k. 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. A vector represented by two different bases (purple and red arrows). ![]() The linear combinations relating the first basis to the other extend to a linear transformation, called the change of basis. ![]() If they are linearly independent, these form a new basis. ![]() 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). A linear combination of one basis of vectors (purple) obtains new vectors (red). For a square matrix, the transpose of the cofactor matrix ![]()
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