Linear Algebra

288 Chapter Four. DeterminantsIDefinitionFor 1×1 matrices, determining nonsingularity is trivial.(a)is nonsingular iff a ≠ 0The 2×2 formula came out in the course of developing the inverse.( )a bis nonsingular iff ad − bc ≠ 0c dThe 3×3 formula can be produced similarly (see Exercise 9).⎛⎝ a b c⎞d e f⎠ is nonsingular iff aei + bfg + cdh − hfa − idb − gec ≠ 0g h iWith these cases in mind, we posit a family of formulas, a, ad−bc, etc. For eachn the formula gives rise to a determinant function det n×n : M n×n → R such thatan n×n matrix T is nonsingular if and only if det n×n (T ) ≠ 0. (We usually omitthe subscript because if T is n×n then ‘det(T )’ could only mean ‘det n×n (T )’.)I.1 ExplorationThis subsection is optional. It briefly describes how an investigator might cometo a good general definition, which is given in the next subsection.The three cases above don’t show an evident pattern to use for the generaln×n formula. We may spot that the 1×1 term a has one letter, that the 2×2terms ad and bc have two letters, and that the 3×3 terms aei, etc., have threeletters. We may also observe that in those terms there is a letter from each rowand column of the matrix, e.g., the letters in the cdh term⎛ ⎞c⎝d⎠hcome one from each row and one from each column. But these observationsperhaps seem more puzzling than enlightening. For instance, we might wonderwhy some of the terms are added while others are subtracted.A good problem solving strategy is to see what properties a solution musthave and then search for something with those properties. So we shall start byasking what properties we require of the formulas.At this point, our primary way to decide whether a matrix is singular isto do Gaussian reduction and then check whether the diagonal of resultingechelon form matrix has any zeroes (that is, to check whether the productdown the diagonal is zero). So, we may expect that the proof that a formula