Principles of Mathematics in Operations Research

52 4 Eigen Values and Vectors

A =

an ai2 ai3

021 <J22 0,23

^31 «32 «33

Volume=det(A)

Fig. 4.1. |det A| = Volume(P).

4.1.2 Properties

1. The determinant of J is 1.
   Example 4.1.8
   10
   0 1 1.


2. The determinant is a linear function of any row, say the first row.
   Example 4.1.9
   det a b
   c d

a b

c d

= ad — cb.

ta tb

c d

= tad — ted = t

a b

c d

3. If A has a zero row, then det .4 = 0.
   Example 4.1.10
   00
   c d
   = 0.


4. The determinant changes sign when two rows are exchanged.
   Example 4.1.11
   c d — cb — ad
   a b

a b

c d

5. The elementary row operations of subtracting a multiple of one row from
   another leaves the determinant unchanged.
   Example 4.1.12

a — acb — ad

c d

(ad — acd) — (be — acd) bc =

a b

c d

6. If two rows are equal (singularity!), then det A = 0.
   Example 4.1.13
   ab

h=o.

a b