Principles of Mathematics in Operations Research

4.1 Determinants 53

7. det AT = det A.
   Example 4.1.14
   a c
   bd

= ad — cb

a b

c d

8. If A is triangular, then det A = Yi7=i a" (det/ = 1).
   Example 4.1.15
   a b
   Od

= ad, «0

c d

= ad.

9. A,B € Rnxn, nonsingular, det(45) = (det A)(det B).
   Example 4.1.16
   a b
   c d

ef

9 h

= (ad — cb)(eh — gf) = adeh — adgf — cbeh + cbgf.

ae + bg af + bh = (ae + bg){cf + dh) - (af + bh)(ce + dg)

ce + dg cf + dh

= aecf + aedh + bgcf + bgdh — afce — afdg — bhce — bhdg

= adeh — adgf — cbeh + cbgf.

10. Let A be nonsingular, A = P-lLDU. Then,
    det A = det P_1 det L det D det U — ±(product of pivots).
    The sign ± is the determinant of P_1 (or P) depending on whether the
    number of row exchanges is even or odd. We know det L = det U = 1 from
    property 7.
    Example 4.1.17 By one Gaussian elimination step, we have
    a b
    c d

1 0

a 1

a 0

n ad—be a

o l

since

a b

c d -»

a b

Od-^

. Thus,

a b

c d

= ad — bc — det D.

11. det A = anAii + aiiAa + ••• + ainAin (property 1!) where A^'s are
    cofactors
    Atj = (-l)i+i det Mij
    where the minor Mij is formed from A by deleting row i and column j.
    Example 4.1.18

an «i2 ai3

O21 022 ^23

0-31 0-32 0-33

an

0,22 023

132 ^33

ai2

^21 a 23

^31 a 33

+

ai3

O21 022

&31 «32

= 011(022033 - 023032) + 012(023031 - 021033) + 013(021032 - o 22 a 3 i)

: 011022033 + 012023031 + 013021032 — 011023032 — 012021033 — 013022031.