Linear Algebra - Free Books

28 Chapter One. Linear SystemsThe Chemistry problem from Example 3.6 is a homogeneous system with morethan one solution so its matrix is singular.⎛⎞7 0 −7 0⎜8 1 −5 −2⎟⎝0 1 −3 0 ⎠0 3 −6 −13.14 Example The first of these matrices is nonsingular while the second issingular ( ) ( )1 2 1 23 4 3 6because the first of these homogeneous systems has a unique solution while thesecond has infinitely many solutions.x + 2y = 03x + 4y = 0x + 2y = 03x + 6y = 0We have made the distinction in the definition because a system (with the samenumber of equations as variables) behaves in one of two ways, depending onwhether its matrix of coefficients is nonsingular or singular. A system wherethe matrix of coefficients is nonsingular has a unique solution for any constantson the right side: for instance, Gauss’ method shows that this systemx + 2y = a3x + 4y = bhas the unique solution x = b − 2a and y = (3a − b)/2. On the other hand, asystem where the matrix of coefficients is singular never has a unique solutions —it has either no solutions or else has infinitely many, as with these.x + 2y = 13x + 6y = 2x + 2y = 13x + 6y = 3Thus, ‘singular’ can be thought of as connoting “troublesome”, or at least “notideal”.The above table has two factors. We have already considered the factoralong the top: we can tell which column a given linear system goes in solely byconsidering the system’s left-hand side — the constants on the right-hand sideplay no role in this factor. The table’s other factor, determining whether aparticular solution exists, is tougher. Consider these two3x + 2y = 53x + 2y = 53x + 2y = 53x + 2y = 4with the same left sides but different right sides. Obviously, the first has asolution while the second does not, so here the constants on the right side