column in which 2 lies, the second term isminusone times the determinant formed
by striking out the row and column in which 1 lies, the third term isplus3 times the
determinant formed by striking out the row and column in which 3 lies, and the
fourth term isminus0 times the determinant formed by striking out the row and
column in which 0 lies; thus starting with the element of row 1, column 1, we move
along the row and multiply byþ1,"1,þ1,"1. It is also possible to start at, say
element (2,1), the number 1, and move across the second row (",þ,",þ), or to
start at element (1,2) and go down the column (",þ,",þ), etc. One would likely
choose to work along a row or column with the most zeroes. The (n"1)&(n"1)
determinants formed in expanding ann&ndeterminant are calledminors, and a
minor with its appropriateþor"sign is acofactor. Expansion of determinants
using minors/cofactors is called Lagrange expansion (Joseph Louis Lagrange
1773). There are also other approaches to expanding determinants, such as manip-
ulating them to make all the elements but one of a row or column zero; see any text
on matrices and determinants. The third-order determinants in the example above
can be reduced to second-order ones and so the fourth-order determinant can be
evaluated as a single number. Obviously every determinant has a corresponding
square matrix and every square matrix has a corresponding determinant, but a
determinant is not a matrix; it is a function of a matrix, a rule that tells us how to
take the set of numbers in a matrix and get a new number. Approaches to the study
of determinants were made by Seki in Japan and Leibnitz in Europe, both in 1683.
The word “determinant” was first used in our sense by Cauchy (1812), who also
wrote the first definitive treatment of the topic.
4.3.3.7 Some Properties of Determinants
These are stated in terms of rows, but also hold for columns;Dis “the determinant”.
- If each element of a row is zero,Dis zero (obvious from Lagrange expansion).
- Multiplying each element of a row bykmultipliesDbyk(obvious from
Lagrange expansion). - Switching two rows changes the sign ofD(since this changes the sign of each
term in the expansion). - If two rows are the sameDis zero. (follows from 3, since ifn¼"n,nmust be
zero. - If the elements of one row are a multiple of those of another,Dis zero (follows
from 2 and 4). - Multiplying a row bykand adding it (adding corresponding elements) to another
row causes no further change inD(in the Laplace expansion the terms withoutk
cancel). - A determinantAcan be written as the sum of two determinantsBandCwhich
differ only in rowiin accordance with this rule: if rowiofAisbi 1 þci 1 bi 2 þci 2
...then rowiofBisbi 1 bi 2 ...and rowiofCisci 1 ci 2 ...An example makes this
clear; with rowi¼row 3:
4.3 The Application of the Schr€odinger Equation to Chemistry by H€uckel 117