Inverse of a Matrix

The following article is from The Great Soviet Encyclopedia (1979). It might be outdated or ideologically biased.

Inverse of a Matrix

For a given square matrix A = ǀǀaijǀǀn1 of order n there exists a matrix B = ǀǀbijǀǀn1 of the same order (called inverse matrix) such that AB = E, where E is the unit matrix; then the equation BA = E also holds. The inverse of a matrix A is designated as A–1. For the existence of the inverse of a matrix A–1, it is necessary and sufficient that the determinant of the given matrix A be nonzero; that is, the matrix A must be nonsingular. The elements bij of the inverse of a matrix are found by the formula bij = Aji/D, where Aji is the cofactor of the element aij of matrix A and D is the determinant of matrix A.

References in periodicals archive ?
Hager  reviewed the formulae proposed by Sherman-Morrison and Woodbury that relate the inverse of a matrix after a small-rank perturbation to the inverse of the original matrix.
Zhao, "An efficient computation of generalized inverse of a matrix," Applied Mathematics and Computation, vol.
Wei, "Iterative methods for the Drazin inverse of a matrix with a complex spectrum," Applied Mathematics and Computation, vol.
This stage is the inverse of a matrix multiplication by constants in the encryption process.
Recall that the Moore-Penrose inverse of a matrix A [member of] [C.sup.mxn] is a matrix X [member of] [C.sup.nxm] which satisfies
Where the matrix O = AT-AT BB+ and (OA) + represent the Moore-Penrose generalized inverse of a matrix (OA).
Schreiber, "An improved Newton iteration for the generalized inverse of a matrix, with applications," SIAM Journal on Scientific and Statistical Computing, vol.
Schreiber, "An improved Newton iteration for the generalized inverse of a matrix, with applications," SIAM: Journal on Scientific and Statistical Computing, vol.
Secondary school teachers may use this activity to consolidate their students' learning of certain concepts of matrices such as the algorithm for matrix multiplication and the concept of the multiplicative inverse of a matrix.
Subroutines for Testing Programs that Computer the Generalized Inverse of a Matrix. 12,3 (Sept.
The case q = -1, which leads to estimates of the trace of the inverse of a matrix, was studied in ; on this problem, see .
Structure of the Inverse of a Matrix. The sparsity structure of the inverse of a square, non-singular matrix is determined by the path structure of the directed graph of the matrix.
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