9.3 DETERMINANTS OF ORDER n
 
We have defined the 2 by 2 determinant

to be the number ad - bc obtained from the square array

of 4 numbers in two rows and two columns. Thus, bordering the array (2) with vertical lines converts the array into a symbol for the number D. Similarly, a 3 by 3 determinant is a number obtained in a certain manner from a square array of 9 numbers.
 
Our next objective is to define an n by n determinant. More precisely, we seek an unambiguous rule for obtaining a number from an n by n square array of numbers and want this rule to agree with the previous definitions when n = 2 or 3. Let
 

 be an array of n² numbers a_ij, where the first subscript designates the row and the second designates the column.
 
There are n! products
 

 with exactly one factor from each row and exactly one factor from each column. The determinant D associated with the array (3) is the sum of the n! terms
 

where the plus sign is used when the permutation
 

is even, and the minus sign is used when the permutation is odd. As before, the array (3) is bordered with vertical lines in writing the symbol
 

 
for a determinant D of order n.

 Wednesday, June 10, 1998