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a3l a32 a33

We may describe this sum as follows. Let Aij be the matrix obtained

from A by deleting the i-th row and the j-th column. Then the sum ex

pressing Det(A) can be written 144

DETERMINANTS

and similarly for the third term. The proof with respect to the other

column is analogous. Furthermore, if t is a number, then

because each 2 x 2 determinant is linear in the first column, and we can

take t outside each one of the second and third terms. Again the proof

is similar with respect to the other columns. A direct substitution shows

that if two adjacent columns are equal, then formula (*) yields 0 for the

determinant. Finally, one sees at once that if A is the unit matrix, then

Det(A) = 1. Thus the three properties are verified.

In the above proof, we see that the properties of 2 x 2 determinants

are used to prove the properties of 3 x 3 determinants. 146

DETERMINANTS

[VI, §2]

Furthermore, there is no particular reason why we selected the expan

sion according to the first row. We can also use the second row, and

write a similar sum, namely:

Again, each term is the product of a2j times the determinant of the 2 x 2

matrix obtained by deleting the second row and j-th column, and putting

the appropriate sign in front of each term. This sign is determined ac

cording to the pattern:

One can see directly that the determinant can be expanded according to

any row by multiplying out all the terms, and expanding the 2 x 2 deter

minants, thus obtaining the determinant as an alternating sum of six