Page 84 - A Course in Linear Algebra with Applications

68 Chapter Three: Determinants

Example 3.1.5
Compute the determinant

1 2 0
4 2 - 1 .
6 2 2

For example, we may expand by row 1, obtaining

2 - 1 3 4 - 1 4 4 2
+ 2 ( - l ) + 0(-l)
2 2 6 2 6 2

= 6 - 28 + 0 = -22.

Alternatively, we could expand by column 2:

4 - 1 4 1 0 5 1 0
+ 2(-l) + 2(-l)
6 2 6 2 4 - 1

= 28 + 4 + 2 = -22.
-
However there is an obvious advantage in expanding by a row
or column which contains as many zeros as possible.
The determinant of a triangular matrix can be written
down at once, an observation which is used frequently in cal-
culating determinants.

Theorem 3.1.5
The determinant of an upper or lower triangular matrix equals
the product of the entries on the principal diagonal of the ma-
trix.

Proof
Suppose that A = (oij) n)Tl is, say, upper triangular, and ex-
pand det(i4) by column 1. The result is the product of a n
and an (n — 1) x (n — 1) determinant which is also upper tri-
angular. Repeat the operation until a 1 x 1 determinant is
obtained (or use mathematical induction).