Linear Algebra

296 Chapter Four. DeterminantsIDefinitionDetermining nonsingularity is trivial for 1×1 matrices.( )a is nonsingular iff a ≠ 0Corollary Three.IV.4.11 gives the formula for the inverse of a 2×2 matrix.( )a bis nonsingular iff ad − bc ≠ 0c dWe can produce the 3×3 formula as we did the prior one, although the computationis intricate (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. Foreach n the formula gives rise to a determinant function det n×n : M n×n → Rsuch that an n×n matrix T is nonsingular if and only if det n×n (T) ≠ 0. (Weusually omit the subscript n×n because the size of T tells us which determinantfunction we mean.)I.1 ExplorationThis is an optional motivation of the general definition, suggesting how aperson might develop that formula. The definition is in the next subsection.Above, in each case the matrix is nonsingular if and only if some formulais nonzero. But the three cases don’t show an obvious pattern for the formula.We may spot that the 1×1 term a has one letter, that the 2×2 terms ad andbc have two letters, and that the 3×3 terms aei, etc., have three letters. Wemay also spot that in those terms there is a letter from each row and column ofthe matrix, e.g., in the cdh term one letter comes from each row and from eachcolumn.⎛ ⎞⎜⎝dhBut these observations are perhaps more puzzling than enlightening. For instance,we might wonder why we add some of the terms while we subtract others.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.c⎟⎠