Linear Algebra

316 Chapter Four. DeterminantsProof. The two statements are equivalent because |t(S)| = |T S|, as both givethe size of the box that is the image of the unit box E n under the compositiont ◦ s (where s is the map represented by S with respect to the standard basis).First consider the case that |T | = 0. A matrix has a zero determinant if andonly if it is not invertible. Observe that if T S is invertible, so that there is anM such that (T S)M = I, then the associative property of matrix multiplicationT (SM) = I shows that T is also invertible (with inverse SM). Therefore, if Tis not invertible then neither is T S — if |T | = 0 then |T S| = 0, and the resultholds in this case.Now consider the case that |T | ̸= 0, that T is nonsingular. Recall that anynonsingular matrix can be factored into a product of elementary matrices, sothat T S = E 1 E 2 · · · E r S. In the rest of this argument, we will verify that if Eis an elementary matrix then |ES| = |E| · |S|. The result will follow becausethen |T S| = |E 1 · · · E r S| = |E 1 | · · · |E r | · |S| = |E 1 · · · E r | · |S| = |T | · |S|.If the elementary matrix E is M i (k) then M i (k)S equals S except that row ihas been multiplied by k. The third property of determinant functions thengives that |M i (k)S| = k · |S|. But |M i (k)| = k, again by the third propertybecause M i (k) is derived from the identity by multiplication of row i by k, andso |ES| = |E| · |S| holds for E = M i (k). The E = P i,j = −1 and E = C i,j (k)checks are similar.QED1.6 Example Application of the map t represented with respect to the standardbases by (1)1−2 0will double sizes of boxes, e.g., from this⃗v⃗w∣ 2 11 2∣ = 3to thist( ⃗w)t(⃗v) ∣ ∣∣∣ 3 3−4 −2∣ = 61.7 Corollary If a matrix is invertible then the determinant of its inverse isthe inverse of its determinant |T −1 | = 1/|T |.Proof. 1 = |I| = |T T −1 | = |T | · |T −1 |QEDRecall that determinants are not additive homomorphisms, det(A + B) neednot equal det(A) + det(B). The above theorem says, in contrast, that determinantsare multiplicative homomorphisms: det(AB) does equal det(A) · det(B).