Linear Algebra

28 Chapter One. Linear SystemsWe 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 where thematrix of coefficients is nonsingular has a unique solution for any constants onthe right side: for instance, Gauss’s 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 solution —it has either no solutions or else has infinitely many, as with these.x + 2y = 13x + 6y = 2x + 2y = 13x + 6y = 3We use the word singular because it means “departing from general expectation”and people often, naively, expect that systems with the same number ofvariables as equations will have a unique solution. Thus, we can think of theword as connoting “troublesome,” or at least “not ideal.” (That ‘singular’ appliesto those systems that do not have one solution is ironic, but it is the standardterm.)3.13 Example The systems from Example 3.3, Example 3.4, and Example 3.8each have an associated homogeneous system with a unique solution. Thus thesematrices are nonsingular.⎛ ⎞ ⎛ ⎞(3 42 −1)⎜3 2 1 ⎟⎝6 −4 0⎠0 1 11 2 −1⎜ ⎟⎝2 4 0⎠0 1 −3The Chemistry problem from Example 3.5 is a homogeneous system with morethan one solution so its matrix is singular.⎛⎞7 0 −7 08 1 −5 −2⎜⎟⎝0 1 −3 0⎠0 3 −6 −1The above table has two dimensions. We have considered the one on top: wecan tell into which column a given linear system goes solely by considering thesystem’s left-hand side — the constants on the right-hand side play no role inthis factor.The table’s other dimension, determining whether a particular solution exists,is tougher. Consider these two3x + 2y = 53x + 2y = 53x + 2y = 53x + 2y = 4