Page 510 - Thomson, William Tyrrell-Theory of Vibration with Applications-Taylor _ Francis (2010)

Appen. C.1 Determinant 497

Example C.1-1
Given the third-order determinant

2 1
4 2
2 0
The minor of the term Uj. = 4 is

1 5
A/o, of 2 1 = 3
2 0 3 0 3

and its cofactor is
Q i = ( “ !)■"'3 = -3

Expansion of a Determinant
The order of a determinant can be reduced by 1 by expanding any row or column

in terms of its cofactors.
Example C.1-2
The determinant of the previous example is expanded in terms of the second column
as
2 1 5
D = 4 2 1 - 1 (-1 ) 4 3 1 + 2(-l)^ 2 5
2
2
3
2 0 3
2 5
+ 0(-l)^
4 1
= -10 - 8 = -18

Properties of Determinants

The following properties of determinants are stated without proof:

1. Interchange of any two columns or rows changes the sign of the determi­
nant.
2. If two rows or two columns are identical, the determinant is zero.
3. Any row or column may be multiplied by a constant and added to another
row or column without changing the value of the determinant.