determinant 127

Solving for xin the first equation and substituting the

result into the second equation yields the solution:

This, of course, is valid only if the quantity ad – bc is

not zero. We call ad – bc the “determinant” of the 2 ×2

matrix of coefficients:

Notice that this determinant is obtained from the

matrix by selecting two elements of the matrix at a

time, one located in each row and column of the

matrix, and assigning a + or – sign according to

whether the order in which the columns are chosen is

an even or an odd

PERMUTATION

:

In the same way, a system of three simultaneous

equations:

ax + by + cz = p

dx + ey + fz = q

gx + hy + iz = r

has a solution provided the quantity aei – afh + bfg –

bdi + cdh – ceg is not zero. We call this quantity the

determinant of the 3 ×3 square matrix:

It is obtained from the matrix by selecting three ele-

ments of the matrix at a time, one located in each row

and column of the matrix, and assigning a + or – sign

according to whether the order in which the columns

are chosen is an even or an odd permutation:

Notice that the signs of the products can equivalently

be evaluated in terms of the sign of row permutations.

In general, the determinant of an n×nmatrix is

formed by selecting nelements of the matrix, arranged

one per row and one per column, forming the product

of those entries, assigning the appropriate sign, and

adding together all possible results. The determinant of

a square matrix Ais denoted det(A) or, sometimes, |A|.

The determinant function satisfies a number of key

properties:

1. If a column or a row of a matrix Ais

entirely zero, then det(A) = 0.

(Each product formed in computing the determinant

will contain a term that is zero.)

2. If two columns or two rows of the matrix

are interchanged, then the sign of det(A)

changes.

(If two columns undergo one more interchange, then the

sign of each permutation alters. Since the process of

forming the determinant can equivalently be viewed in

terms of row permutations, the same is true if two rows

are interchanged.)

3. If a matrix Ahas two identical columns, or

two identical rows, then det(A) = 0.

(Interchange those two columns or two rows. The

matrix remains unchanged, yet the determinant has

opposite sign. It must be the case then that det(A) = 0.)