Cramer's Rule and Inverse Matrix

Linear Algebra and its Applications

In this paper, we obtain an explicit representation of the {2}-inverse A (2) T ,S of a matrix A ∈ C m×n with the prescribed range T and null space S. As special cases, new expressions for the Moore-Penrose inverse A + and Drazin inverse A D are derived. Through explicit expressions, we re-derive the condensed Cramer rules of Werner for minimal-norm least squares solution of linear equations Ax = b and propose two new condensed Cramer rules for the unique solution of a class of singular system Ax = b, x ∈ R(A k ), b ∈ R(A k ), k = Ind(A). Finally, condensed determinantal expressions for A + , A D , AA + , A + A, and AA D are also presented.

2015

By a generalized inverse of a given matrix, we mean a matrix that exists for a larger class of matrices than the nonsingular matrices, that has some of the properties of the usual inverse, and that agrees with inverse when given matrix happens to be nonsingular. In theory, there are many different generalized inverses that exist. We shall consider the Moore Penrose, weighted Moore-Penrose, Drazin and weighted Drazin inverses. New determinantal representations of these generalized inverse based on their limit representations are introduced in this paper. Application of this new method allows us to obtain analogues classical adjoint matrix. Using the obtained analogues of the adjoint matrix, we get Cramer's rules for the least squares solution with the minimum norm and for the Drazin inverse solution of singular linear systems. Cramer's rules for the minimum norm least squares solutions and the Drazin inverse solutions of the matrix equations AX = D, XB = D and AXB = D are also obtained, where A, B can be singular matrices of appropriate size. Finally, we derive determinantal representations of solutions of the differential matrix equations, X ′ + AX = B and X ′ + XA = B, where the matrix A is singular.

Linear and Multilinear Algebra, 2008

In this article, we introduce determinantal representations of the Moore -Penrose inverse and the Drazin inverse which are based on analogues of the classical adjoint matrix. Using the obtained analogues of the adjoint matrix, we get Cramer rules for the least squares solution and for the Drazin inverse solution of singular linear systems. Finally, determinantal expressions for A + A, AA + , and A D A are presented.

International Journal of Technology Diffusion, 2010

In this paper, a new algorithm is proposed for finding inverse and determinant of a given matrix in one instance. The algorithm is straightforward in understanding and manual calculations. Computer implementation of the algorithm is extremely simple and is quite efficient in time and memory utilization. The algorithm is supported by an example. The number of multiplication/division performed by the algorithm is exactly; however, its efficiency lies in the simplicity of coding and minimal utilization of memory. Simple applicability and reduced execution time of the method is validated form the numerical experiments performed on test problems. The algorithm is applicable in the cases of pseudo inverses for non-square matrices and solution of system of linear equations with minor modification.

Asian Research Journal of Mathematics

Computing the inverse of 3 x 3 square matrices using known methods such as Gauss-Jordon Method needs more time. During examination, candidates need to go through long process to find the inverse of 3 x 3 square matrices. Some students also find it very difficult and confusing when using Gauss-Jordon and Adjugate methods to find the inverse of the matrix. In this note, we present a new method that is simple and easy way of finding the inverse of 3 x 3 square matrices. We further give some applications of matrices in the real world phenomenon. Many other challenging problems can be addressed surprisingly by applying this strategy.