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

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