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Let R be a local ring. There is a determinant map from the matrix ring GL(R ) to the abelianised unit group R × ab with the following properties: [1] The determinant is invariant under elementary row operations; The determinant of the identity matrix is 1; If a row is left multiplied by a in R × then the determinant is left multiplied by a
The set M(n, R) (also denoted M n (R) [7]) of all square n-by-n matrices over R is a ring called matrix ring, isomorphic to the endomorphism ring of the left R-module R n. [58] If the ring R is commutative, that is, its multiplication is commutative, then the ring M(n, R) is also an associative algebra over R.
In business and project management, a responsibility assignment matrix [1] (RAM), also known as RACI matrix [2] (/ ˈ r eɪ s i /; responsible, accountable, consulted, and informed) [3] [4] or linear responsibility chart [5] (LRC), is a model that describes the participation by various roles in completing tasks or deliverables [4] for a project or business process.
In algebra, the Wedderburn–Artin theorem is a classification theorem for semisimple rings and semisimple algebras.The theorem states that an (Artinian) [a] semisimple ring R is isomorphic to a product of finitely many n i-by-n i matrix rings over division rings D i, for some integers n i, both of which are uniquely determined up to permutation of the index i.
The matrix ring M n (R) can be identified with the ring of endomorphisms of the free right R-module of rank n; that is, M n (R) ≅ End R (R n). Matrix multiplication corresponds to composition of endomorphisms. The ring M n (D) over a division ring D is an Artinian simple ring, a special type of semisimple ring.
If Gaussian elimination applied to a square matrix A produces a row echelon matrix B, let d be the product of the scalars by which the determinant has been multiplied, using the above rules. Then the determinant of A is the quotient by d of the product of the elements of the diagonal of B : det ( A ) = ∏ diag ( B ) d . {\displaystyle \det ...
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In other words, the matrix of the combined transformation A followed by B is simply the product of the individual matrices. When A is an invertible matrix there is a matrix A −1 that represents a transformation that "undoes" A since its composition with A is the identity matrix. In some practical applications, inversion can be computed using ...