Linear Algebra

322 Chapter Four. DeterminantsThe sign returned by the size function reflects the orientation or sense of thebox. (We see the same thing if we picture the effect of scalar multiplication by anegative scalar.)Although it is both interesting and important, we don’t need the idea oforientation for the development below and so we will pass it by. (See Exercise 27.)1.3 Definition The volume of a box is the absolute value of the determinant of amatrix with those vectors as columns.1.4 Example By the formula that takes the area of the base times the height, thevolume of this parallelepiped is 12. That agrees with the determinant.⎛ ⎞2⎝0⎠2⎛ ⎞0⎝3⎠1⎛ ⎞−1⎝ 0 ⎠12 0 −10 3 0= 12∣2 1 1∣We can also compute the volume as the absolute value of this determinant.0 2 03 0 3= −12∣1 2 1∣The next result describes some of the geometry of the linear functions thatact on R n .1.5 Theorem A transformation t: R n → R n changes the size of all boxes by thesame factor, namely the size of the image of a box |t(S)| is |T| times the size ofthe box |S|, where T is the matrix representing t with respect to the standardbasis.That is, for all n×n matrices, the determinant of a product is the productof the determinants |TS| = |T| · |S|.The two sentences say the same thing, first in map terms and then in matrixterms. This is because |t(S)| = |TS|, as both give the size of the box that isthe image of the unit box E n under the composition t ◦ s (where s is the maprepresented by S with respect to the standard basis).Proof First consider the |T| = 0 case. A matrix has a zero determinant if andonly if it is not invertible. Observe that if TS is invertible then there is an Msuch that (TS)M = I, so T(SM) = I, which shows that T is invertible, withinverse SM. By contrapositive, if T is not invertible then neither is TS — if|T| = 0 then |TS| = 0.Now consider the case that |T| ≠ 0, that T is nonsingular. Recall that anynonsingular matrix factors into a product of elementary matrices T = E 1 E 2 · · · E r .