For a square matrix, the transpose of the cofactor matrix
In linear algebra, the adjugate of a square matrix A is the transpose of its cofactor matrix and is denoted by adj(A).[1][2] It is also occasionally known as adjunct matrix,[3][4] or "adjoint",[5] 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.
^Gantmacher, F. R. (1960). The Theory of Matrices. Vol. 1. New York: Chelsea. pp. 76–89. ISBN 0-8218-1376-5.
^Strang, Gilbert (1988). "Section 4.4: Applications of determinants". Linear Algebra and its Applications (3rd ed.). Harcourt Brace Jovanovich. pp. 231–232. ISBN 0-15-551005-3.
^Claeyssen, J.C.R. (1990). "On predicting the response of non-conservative linear vibrating systems by using dynamical matrix solutions". Journal of Sound and Vibration. 140 (1): 73–84. Bibcode:1990JSV...140...73C. doi:10.1016/0022-460X(90)90907-H.
^Chen, W.; Chen, W.; Chen, Y.J. (2004). "A characteristic matrix approach for analyzing resonant ring lattice devices". IEEE Photonics Technology Letters. 16 (2): 458–460. Bibcode:2004IPTL...16..458C. doi:10.1109/LPT.2003.823104.
^Householder, Alston S. (2006). The Theory of Matrices in Numerical Analysis. Dover Books on Mathematics. pp. 166–168. ISBN 0-486-44972-6.
the adjugate 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...
formula for the inverse involving the adjugatematrix mentioned below. In this event, the determinant of the inverse matrix is given by det ( A − 1 ) = 1 det...
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two vectors u and v. The theorem can also be stated in terms of the adjugatematrix of A: det ( A + u v T ) = det ( A ) + v T a d j ( A ) u , {\displaystyle...
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First, we can directly compute the matrix product M M ′ {\displaystyle MM'} (using simple properties of the adjugatematrix, or alternatively using the formula...
and dyadic products can all be expressed in matrix form. Dyadic expressions may closely resemble the matrix equivalents. The dot product of a dyadic with...
case this is an elementary fact because the inverse of a matrix is given as the adjugatematrix divided by its determinant). The method of proof here can...
matrix Jordan normal form Weyr canonical form Rank Matrix inversion, invertible matrix Pseudoinverse Adjugate Transpose Dot product Symmetric matrix Orthogonal...
where Madj is adjugatematrix given by the standard formula - its (i,j)-th element is the column-determinant of the (n − 1) × (n − 1) matrix that results...
an) is the product a1⋯an. The adjugate of a diagonal matrix is again diagonal. Where all matrices are square, A matrix is diagonal if and only if it is...
matrices is regular and the formula for the inverse in terms of the adjugatematrix shows that inversion is regular as well on matrices with determinant...
that for very small problems it is beneficial to replace the matrix inverse with the adjugate, which will yield the same iteration because it is equal to...
B, matrix B is already in Hermite Normal Form so the first step is not needed. The determinant is d = 2 {\displaystyle d=2} , the adjugatematrix A =...
nonsingular in the lower half plane. The analyticity follows from the adjugatematrix formula (since both the entries of Q(t){\displaystyle Q(t)} and...
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