Some Properties of the Determinant Function
The determinant of a
square
matrix
A
, det
A
, is a real number.
However, rather than thinking of the determinant as a
function of the entire matrix
A
it’s nature is better revealed by regarding it as a function of the row vectors
1
that make
up the matrix
A
. The determinant function can then be viewed as having for its domain the
n
fold product
R
n
× · · · ×
R
n
(
n
rows of an
n
×
n
matrix) and range the real numbers
R
. More briefly, det :
R
n
× · · · ×
R
n
→
R
.
If
v
1
, . . . , v
n
are
n
vectors from
R
n
we’ll use either of the notations det(
v
1
, . . . , v
n
) or det
v
1
.
.
.
v
n
to denote the same number.
Here are some properties, characterizing the determinate function. Assume that
v
1
, . . . , v
n
are
n
vectors from
R
n
.
1. For any
i
= 1
, . . . , n
and
c
∈
R
det
v
1
.
.
.
cv
i
.
.
.
v
n
=
c
det
v
1
.
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 Spring '09
 Linear Algebra, Invertible matrix, Det

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