The following theorem is a list of equivalences. We give two proof. If Adoes not have an inverse, Ais called singular. Applications and properties. Say if these statements are true or false. On the other hand, we show that A+I, A-I are nonsingular matrices. So a non singular matrix "must" not have an inverse matrix. If A is non-singular then, a) The last column vector of A, can be written as a linear combination of the first three column vectors of A. b) The nullity of A is positive. We show that a nilpotent matrix A is singular. Square matrices that are nonsingular have a long list of interesting properties, which we will start to catalog in the following, recurring, theorem. Invertible Matrices De nitions Facts Properties of Inverses Algorithms for Computing Inverses The Augmentation Method Elementary Matrices The EA = rref(A) Method Linear Algebra in a Nutshell Invertible Means Nonsingular Partial Statement. One uses eigenvalues method Of course, singular matrices will then have all of the opposite properties. This can be proved, for strictly diagonal dominant matrices, using the Gershgorin circle theorem. NON{SINGULAR MATRICES DEFINITION. This result is known as the Levy–Desplanques theorem. Properties of Inverse Matrices: If A is nonsingular, then so is A-1 and (A-1) -1 = A If A and B are nonsingular matrices, then AB is nonsingular and (AB)-1 = B-1 A-1 If A is nonsingular then (A T)-1 = (A-1) T If A and B are matrices with AB=I n then A and B are inverses of each other. (Non{singular matrix) An n n Ais called non{singular or invertible if there exists an n nmatrix Bsuch that AB= In= BA: Any matrix Bwith the above property is called an inverse of A. THEOREM. (Inverses are unique) If Ahas inverses Band C, then B= C. If Ahas an inverse, it is denoted by A 1. M-Matrix Characterizations.l-Nonsingular M-Matrices R. J. Plemmons* Departments of Computer Science and Mathenuitics University of Tennessee Knoxville, Tennessee 37919 Submitted by Hans Schneider ABSTRACT The purpose of this survey is to classify systematically a widely ranging list of characterizations of nonsingular M-matrices from the economics and mathematics literatures. Provide an explanation as to why they are that way. A strictly diagonally dominant matrix (or an irreducibly diagonally dominant matrix) is non-singular.
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