Page 192 - 35Linear Algebra

192 Determinants

Example 108 Continuing with the previous example,

   
3 −1 −1 2 0 2
adj  1 2 0  =  −1 3 −1  .
0 1 1 1 −3 7

Now, multiply:

     
3 −1 −1 2 0 2 6 0 0
1 2 0 −1 3 −1 = 0 6 0
     
0 1 1 1 −3 7 0 0 6
−1
   
3 −1 −1 2 0 2
1
⇒  1 2 0  =  −1 3 −1 
6
0 1 1 1 −3 7
This process for finding the inverse matrix is sometimes called Cramer’s Rule .

8.4.3 Application: Volume of a Parallelepiped

3
Given three vectors u, v, w in R , the parallelepiped determined by the three
vectors is the “squished” box whose edges are parallel to u, v, and w as
depicted in Figure 8.8.
You probably learnt in a calculus course that the volume of this object is
|u (v × w)|. This is the same as expansion by minors of the matrix whose
columns are u, v, w. Then:

Volume = det u v w

192