Linear Algebra

Section II. Geometry of Determinants 323To finish this argument we will verify that |ES| = |E| · |S| for all matrices S andelementary matrices E. The result will then follow because |TS| = |E 1 · · · E r S| =|E 1 | · · · |E r | · |S| = |E 1 · · · E r | · |S| = |T| · |S|.There are three kinds of elementary matrix. We will cover the M i (k) case;the P i,j and C i,j (k) checks are similar. We have that M i (k)S equals S exceptthat row i is multiplied by k. The third property of determinant functionsthen gives 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. Thus|ES| = |E| · |S| holds for E = M i (k).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 thisto this⃗v⃗w2 1∣1 2∣ = 3t( ⃗w)∣ t(⃗v) ∣∣∣∣3 3−4 −2∣ = 61.7 Corollary If a matrix is invertible then the determinant of its inverse is theinverse of its determinant |T −1 | = 1/|T|.Proof 1 = |I| = |TT −1 | = |T| · |T −1 |QEDRecall that determinants are not additive homomorphisms, that det(A + B)need not equal det(A) + det(B). In contrast, the above theorem says thatdeterminants are multiplicative homomorphisms: det(AB) equals det(A)·det(B).Exercises1.8 Find(the(volume of the region defined by the vectors.1 −1(a) 〈 , 〉3)4)⎛ ⎞ ⎛ ⎞ ⎛ ⎞2 3 8(b) 〈 ⎝1⎠ , ⎝−2⎠ , ⎝−3⎠〉0 4 8⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞1 2 −1 0(c) 〈 ⎜2⎟⎝0⎠ , ⎜2⎟⎝2⎠ , ⎜ 3⎟⎝ 0⎠ , ⎜1⎟⎝0⎠ 〉1 2 5 7