If Adoes not have an inverse, Ais called singular. (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. The following theorem is a list of equivalences. 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. Of course, singular matrices will then have all of the opposite properties. We give two proof. Applications and properties. One uses eigenvalues method Say if these statements are true or false. We show that a nilpotent matrix A is singular. Provide an explanation as to why they are that way. This result is known as the Levy–Desplanques theorem. Square matrices that are nonsingular have a long list of interesting properties, which we will start to catalog in the following, recurring, theorem. So a non singular matrix "must" not have an inverse matrix. (Inverses are unique) If Ahas inverses Band C, then B= C. If Ahas an inverse, it is denoted by A 1. 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. A strictly diagonally dominant matrix (or an irreducibly diagonally dominant matrix) is non-singular. This can be proved, for strictly diagonal dominant matrices, using the Gershgorin circle 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. 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. NON{SINGULAR MATRICES DEFINITION. THEOREM. On the other hand, we show that A+I, A-I are nonsingular matrices.

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