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Determinants, Matrix Norms, Inverse Mapping Theorem
G. B. Folland
The purpose of this notes is to present some useful facts about matrices and determinants
and a proof of the inverse mapping theorem that is rather diﬀerent from the one in Apostol.
Notation:
M
n
(
R
) denotes the set of all
n
×
n
real matrices.
Determinants
: If
A
∈
M
n
(
R
), we can consider the rows of
A
:
r
1
,...,
r
n
. These are
elements of
R
n
, considered as row vectors. Conversely, given
n
row vectors
r
1
,...,
r
n
, we can
stack them up into an
n
×
n
matrix
A
. Thus we can think of a function
f
on matrix space
M
n
(
R
) as a function of
n
R
n
valued variables or vice versa:
f
(
A
)
←→
f
(
r
1
,...,
r
n
)
.
Basic Fact
:
There is a unique function
det :
M
n
(
R
)
→
R
(the
“determinant”
) with the
following three properties:
i. det
is a linear function of each row when the other rows are held ﬁxed: that is,
det(
α
a
+
β
b
,
r
2
,...,
r
n
) =
α
det(
a
,
r
2
,...,
r
n
) +
β
det(
b
,
r
2
,...,
r
n
)
,
and likewise for the other rows.
ii.
If two rows of
A
are interchanged,
det
A
is multiplied by

1
:
det(
...,
r
i
,...,
r
j
,...
) =

det(
...,
r
j
,...,
r
i
,...
)
.
iii. det(
I
) = 1,
where
I
denotes the
n
×
n
identity matrix.
The uniqueness of the determinant follows from the discussion below; existence takes more
work to establish. We shall not present the proof here but give the formulas. For
n
= 2 and
n
= 3 we have
det
±
a b
c d
²
=
ad

bc,
det
a b c
d e f
g h i
=
aei

a
fh
+
b
fg

bdi
+
cdh

ceg.
For general
n
, det
A
=
∑
σ
(sgn
σ
)
A
1
σ
(1)
A
2
σ
(2)
···
A
nσ
(
n
)
, where the sum is over all permuta
tions
σ
of
{
1
,...,n
}
, and sgn
σ
is 1 or

1 depending on whether
σ
is obtained by an even
or odd number of interchanges of two numbers. (This formula is a computational nightmare
for large
n
, being a sum of
n
! terms, so it is of little use in practice. There are better ways
to compute determinants, as we shall see shortly.)
An important consequence of properties (i) and (ii) is
iv.
If one row of
A
is the zero vector, or if two rows of
A
are equal, then
det
A
= 0
.
Properties (i), (ii), and (iv) tell how the determinant of a matrix behaves under the
elementary row operations:
1
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View Full Document – Multiplying a row by a scalar multiplies the determinant by that scalar.
– Interchanging two rows multiplies the determinant by

1.
– Adding a multiple of one row to another row leaves the determinant unchanged, because
det(
...,
r
i
+
α
r
j
,...,
r
j
,...
) = det(
...,
r
i
,...,
r
j
,...
) +
α
det(
...,
r
j
,...,
r
j
,...
)
,
by (i), and the last term is zero by (iv).
This gives a reasonably eﬃcient way to compute determinants. To wit, any
n
×
n
matrix
A
can be rowreduced either to a matrix with an allzero row (whose determinant is 0) or
to the identity matrix (whose determinant is 1). Just keep track of what happens to the
determinant as you perform these row operations, and you will have calculated det
A
. (There
are shortcuts for this procedure, but that’s another story. We’ll mostly be dealing with 2
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This note was uploaded on 01/14/2011 for the course ECE 210a taught by Professor Chandrasekara during the Fall '08 term at UCSB.
 Fall '08
 Chandrasekara

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