Identity
Matrix
A square matrix that is zero everywhere other tha
the main diagonal and I on the main diagonal
o
di
is
denoted
0
O
O O 1
I
by
I
Id
In
Iymatrix
n xn
9
A
912 9
92 922 923
9
911 912 913
912 913
92 922923
92 922 923
2
I
G
2 2
acts Ii
a
9
3
92 922 923
2
3
13
3
i
91 912 913
92 azz 923
For A
an
If
R
an
n xn
Im A
A
reduced row echelon form of
matrix A then either R has
is the
of zeros
row
Every
occur
move
or
É
row
has
a
the identity
is
or
progressively to the
down the matrix
is
the
Entirely'tsieros
the last row
Leading 1
is
matrix I
it does not
leading
I
every leading
R
R
this
f
Rs
and
1.4.3
Theorem
a
A
A In
man
on
1
has a
but leading 1s
right
the
identity matrix
as
main
we
diagonal
of
Inverse
I at
Y a e IR
so
we
matrix
an
want
an
such that
analogy
I
for an
there exist
a
d
U
in
a
i
a
1
matrices
Definition 2
A is a square matrix and if a matrix
B of the same size can be found such
I
BA
then A is said to be
that AB
invertible and B is called an Inverse of A
if no such matrix B can be found then A
If
is said to
A
be
3
singular
3
Be
AB
I 3
I
BA
I
3
EE
o
3
9
9
BA
AB
Theorem
If
1.4.4
B and C
then
B
I
C
B
AB
BA
AC
CA
is
Be C
I
of A
BA C
I
BA
IC
AC
IC
C
C
B CI
BA C
C
I
inverse
an
B
both inverses of matrix A
are
B
B AC
BI
B
makes
It
inverse of
If
A is
but
in
invertible
we
denote it's
by A
arithmetic
At
Theorem
the
A
inverse
But
to talk about
sense
a
e
ta
a
not in
matrices
I
4
1
1.4.5
The matrix
EI
A
it and only if
in which case the inverse
is inversible
the formula
A
ad
is
be
0
given by
d
age f y
c
a
by
ax
dy
Cx
u
V
Y
E
E
If
invertible
is
to solve this
way
demonstration that
EI
91
89
y
I
I
y
so
we
will
we
have
assume
for
invertible
it is
J
5
55
1
a simp
E
fall
x
a
d
I
Theorem 1.4.6
If
A and B
AB
then
same size
invertible matrices with
are
ABI
AB
is
invertible and
B A
A
B A
BB
A
AI A
AA
I
B A
B CA A B
AB
BIB
AB CA BI
BA
is
CA
the
BY
inverse
I
B B
AB
I
of AB
the
number of invertible matrices
A product of
any
the product is
is invertible and the inverse of
the product of the
Powers of
A is
If
A
If
A
a
s
in
reverse
order
matrix
square matrix the
A
I
A
a
inverses
is
invertible
A
At
The usual laws of
we
A
non
define
A Cn times
AAA
s
we
define
G
times
negative exponents
for example
Ar As Arts Cars
Ars
apply
Theorem
If
1.4.7
A is invertible and
then
is
n
a nonnegative
integer
invertible and CA 5
A
A
is
B
A
is invertible and
C KA is
scalar
Proof
invertible
K
KA
and
for
A
Ant
K A
1
I
I
K A
KA
K'K A
II
l
AA
Kk
K A
zero
non
any
KA
A
A
I
A l
Theorem
It
1.4.8
the sizes
of the matrix
stated operations
A
ATT
B
LA BIT
such that the
be performed then
can
A
Att
B
BIT AT AB
C
A
D
KAT
KAT
E
ABIT
BTAT
The transpose of
of matrices
in the
are
is
a
product of
any
number
the product of the transposes
reverse order
1.4.9
Theorem
A
If
is
an
invertible matrix then
invertible and
also
CATS
AT CA TT
At
A
A
I
CAST
A
A
IT AT
AJ
IT AT AA'T
I
IT
IT
I
I
At
is
1.5
Section
Elementary matrices and
Multiply
a
2
Interchange
3
Add
by
row
1
multiply
2
swap back
if
then
by
A
do
one
row
to another
B
A
Rj
Rj
I
B resulted from
we
constant
rows
Backwards
3
a
constant c times
a
method for
l
finding A
1
a
Rj Rj
CR
CR
s
R2 BR
Re
85
R2
R2 381
s
Definition 1
Matrices
A and B
are
equivalent if either
from the
row
other
operations
A
p
B
by
A
can
a
is
said to b row
be obtained
sequence
now
of
elementary
equivalent to B
Definition
2
matrix
A matrix F is called an elementry
if it can be obtained from an identity
matrix
I
by
performing
Is
Ra
8
It
single
o
3 Rz
Ra
a
Rs
1
g
row
operation
1.5 1
theorem
row operations
results from
If the elementry matrix E
if
A is
EA
is
same
I
certain
a
performing
a
m x n
now
by matrix multiplication
operation Im and
matrix then the product
when this
the matrix that results
row operation is performed on A
RER
EA
ED
E
A
ER
B