NOTES ON SECTION
3
.
2
MA265, SECTIONS 41, 52
Key words
Reduction to triangular form
The formula we developed in the last section is horrendous to use. We developed it so that we would have
an object called “the determinant”; we will now show that the determinant has nice
properties
which we
can use in practice (instead of the definition).
Theorem 1.
The determinant is the unique function from square matrices to the real numbers satisfying
the following
4
properties:
(1)
It is additive in each column (that is, if you have a column
a
+
b
, then the determinant is the sum
of the determinants of the matrices you obtain by replacing that column with the columns
a
and
b
instead).
(2)
It is multiplicative in each column (that is, if you have a column
k
a
, where
k
is a scalar and
a
is a
column vector, then the determinant of the matrix is
k
times the determinant of the matrix you get
by replacing the column
k
a
with
a
).
(3)
It is alternating, meaning that if you exchange two columns then you change the sign of the deter
minant.
(4)
The determinant of the identity matrix is
1
.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
This is the end of the preview.
Sign up
to
access the rest of the document.
 Spring '08
 Bens
 Linear Algebra, Algebra, Determinant, Det

Click to edit the document details