Q1 2) In which of the four cases is the coefficient matrix A of the system non-singular? In which cases is it singular? (Look at the definition below the middle table at the bottom of the page (definition 3.11)) PROOP We've seen examples of all three happening so we need only prove thatthere are no other possibilities.First observe a homogeneous system with at least one non-o solution v hasinfinitely many solutions. This is because any scalar multiple of V also solves thehomogeneous system and there are infinitely many vectors in the set of scalarmultiples of V: if s, t R are unequal then sv tv, since sv-tv = (st)v isnon-Ō as any non-0 component of V, when rescaled by the non-0 factor s-t, willgive a non-0 value.Now apply Lemma 3.7 to conclude that a solution set{p+hh solves the associated homogeneous system)is either empty (if there is no particular solution p), or has one element (if thereis a p and the homogeneous system has the unique solution 0), or is infinite (ifthere is a p and the homogeneous system has a non-o solution, and thus by theprior paragraph has infinitely many solutions).QEDThis table summarizes the factors affecting the size of a general solution.number of solutions of thehomogeneous systemoneinfinitely manyinfinitely manyparticular yessolutionuniquesolutionsolutionsnoexists?nonosolutionssolutionsThe dimension on the top of the table is the simpler one. When we performGauss's Method on a linear system, ignoring the constants on the right side andso paying attention only to the coefficients on the left-hand side, we either endwith every variable leading some row or else we find some variable that does notlead a row, that is, we find some variable that is free. (We formalize "ignoringthe constants on the right" by considering the associated homogeneous system.)A notable special case is systems having the same number of equations asunknowns. Such a system will have a solution, and that solution will be unique,if and only if it reduces to an echelon form system where every variable leads itsrow (since there are the same number of variables as rows), which will happen ifand only if the associated homogeneous system has a unique solution.3.11 Definition A square matrix is nonsingular if it is the matrix of coefficientsof a homogeneous system with a unique solution. It is singular otherwise, thatis, if it is the matrix of coefficients of a homogeneous system with infinitely manysolutions.

In the given problem the concept of rank of a matrix is used to obtained the cases for the existence…

3.11 Definition A square matrix is nonsingular if it is the matrix of coefficients of a homogeneous system with a unique solution. It is singular otherwise, that is, if it is the matrix of coefficients of a homogeneous system with infinitely many solutions.

3.11 Definition A square matrix is nonsingular if it is the matrix of coefficients of a homogeneous system with a unique solution. It is singular otherwise, that is, if it is the matrix of coefficients of a homogeneous system with infinitely many solutions.

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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