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In linear algebra, the adjugate or classical adjoint of a square matrix A, adj(A), is the transpose of its cofactor matrix. [1] [2] It is occasionally known as adjunct matrix, [3] [4] or "adjoint", [5] though that normally refers to a different concept, the adjoint operator which for a matrix is the conjugate transpose.
In linear algebra, the Laplace expansion, named after Pierre-Simon Laplace, also called cofactor expansion, is an expression of the determinant of an n × n-matrix B as a weighted sum of minors, which are the determinants of some (n − 1) × (n − 1)-submatrices of B.
In linear algebra, a minor of a matrix A is the determinant of some smaller square matrix generated from A by removing one or more of its rows and columns. Minors obtained by removing just one row and one column from square matrices (first minors) are required for calculating matrix cofactors, which are useful for computing both the determinant and inverse of square matrices.
Transpose of the cofactor matrix: The inverse of a matrix is its adjugate matrix divided by its determinant: Augmented matrix: Matrix whose rows are concatenations of the rows of two smaller matrices: Used for performing the same row operations on two matrices Bézout matrix: Square matrix whose determinant is the resultant of two polynomials
In matrix theory, the rule of Sarrus is a mnemonic device for computing the determinant of a matrix named after the French mathematician Pierre Frédéric Sarrus. [ 1 ] Consider a 3 × 3 {\displaystyle 3\times 3} matrix
In matrix calculus, Jacobi's formula expresses the derivative of the determinant of a matrix A in terms of the adjugate of A and the derivative of A. [1] If A is a differentiable map from the real numbers to n × n matrices, then
For example, if A is a 3-by-0 matrix and B is a 0-by-3 matrix, then AB is the 3-by-3 zero matrix corresponding to the null map from a 3-dimensional space V to itself, while BA is a 0-by-0 matrix. There is no common notation for empty matrices, but most computer algebra systems allow creating and computing with them.
A real tensor in 3D (i.e., one with a 3x3 component matrix) has as many as six independent invariants, three being the invariants of its symmetric part and three characterizing the orientation of the axial vector of the skew-symmetric part relative to the principal directions of the symmetric part.