Mathematical Methods for Physicists: A concise introduction - Site Map

EXPANSION OF DETERMINANTS
a 11 a 12 a 1n
a 21 a 22 a 2n
D ˆ
. . .
: …A2:8†
. . .
a n1 a n2 a nn
By deleting the ith row and the kth column from the determinant D we obtain
an (n 1)st order determinant (a square array of n 1 rows and n 1 columns
between vertical bars), which is called the minor of the element a ik (which belongs
to the deleted row and column) and is denoted by M ik . The minor M ik multiplied
by …† i‡k is called the cofactor of a ik and is denoted by C ik :
C ik ˆ…1† i‡k M ik :
…A2:9†
For example, in the determinant
a 11 a 12 a 13
a 21 a 22 a 23
;
a 31 a 32 a 33
we have
C 11 ˆ…1† 1‡1 M 11 ˆ a22 a 23
a 32 a 33 ; C 32 ˆ…1† 3‡2 M 32 ˆ a 11 a 13
a 21 a 23 ;
etc:
It is very convenient to get the proper sign (plus or minus) for the cofactor
…1† i‡k by thinking of a checkerboard of plus and minus signs like this
‡ ‡
‡ ‡
‡ ‡ etc:
‡ ‡
.
etc: . .
‡
‡
thus, for the element a 23 we can see that the checkerboard sign is minus.
Expansion of determinants
Now we can see how to ®nd the value of a determinant: multiply each of one row
(or one column) by its cofactor and then add the results, that is,
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