Algebra of Matrices
Real Numbers
(Solve for x)
m n Matrices
(Solve for X)
x + a = b
X + A = B
x + a + (-a) = b + (-a)
X + A + (-A) = B + (-A)
x + 0 = b – a
X + 0 = B - A
x = b – a
X = B - A
matrix
equation
Algebra of Matrices
Real Numbers
(Solve for x)
m n Matrices
(Solve for X)
x + a = b
X + A = B
x + a + (-a) = b + (-a)
X + A + (-A) = B + (-A)
x + 0 = b – a
X + 0 = B - A
x = b – a
X = B - A
matrix
equation
Algebra of Matrices
Real Numbers
(Solve for x)
m n Matrices
(Solve for X)
x
a = b
X A = B
x
a
a
-1
= b
a
-1
X A A
-1
= B A
-1
x = b
a
-1
X = X E = B A
-1
matrix
equation
/ A
-1
R
A X = B
A
-1
A X = A
-1
B
X = E X= A
-1
B
/ A
-1
L
A
B
X
!!!
Definition 7.3:
Definition 7.3:
A square matrix A is called noninvertible (or
singular) when det(A)=0. W przeciwnym
wypadku, matrix A is called nonsingular
(invertible).
Definition 8.3:
Definition 8.3:
An quadratic matrix A (of order n) is nonsingular
(invertible) if there exist a quadratic matrix B
such that
AB = BA = E
n
where E
n
is the identity matrix of order n. The
matrix B is called the (multiplicative) inverse of
A.
!!!
• Nonsquare Matrix DO NOT HAVE inverses
A
nm
· B
mn
B
mn
·
A
nm
quadratic Matrix
of order n
quadratic matrix
of order m
• Not all square matrices possess inverses
V. Matrices – The Inverse of a Matrix
Theorem 1.
Theorem 1.
5
5
:
:
If A is an invertible matrix, then its inverse is
unique. We denote the inverse of A by A
-1
.
Uniqueness of an Inverse Matrix
Proof:
A is invertible it has at least one inverse
Suppose that B and C are inverses of A
AB = BA = E AC = CA = E
AB = E
C(AB) = CE
(CA)B = C
EB = C
B = C= A
-1
How to Find an Inverse Matrix ?
by Gauss-Jordan
Elimination
(elementary row
operations are used)
by Its Adjoint
(determinants
are applied)
Finding the Inverse of Matrix
Example
:
Show that B is the inverse of A :
,
1
1
2
1
,
1
1
2
1
B
A
1
0
0
1
1
2
1
1
2
2
2
1
1
1
2
1
1
1
2
1
AB
1
0
0
1
1
2
1
1
2
2
2
1
1
1
2
1
1
1
2
1
BA
1
A
B
!!!
Finding the Inverse of Matrix by Gauss-Jordan Elimination
E
A
1
A
E
Elementary row
operations
NOTE:
If A cannot be row reduced to E, then A is singular
Example
:
Find the inverse of the following matrix
3
2
6
1
0
1
0
1
1
A
1
0
0
0
1
0
0
0
1
3
2
6
1
0
1
0
1
1
E
A
•
write the n 2n matrix that consists of given matrix A
on the left and the n n identity matrix E on the right to
obtain [A : E]. We call this process adjoining the
matrices A and E.
•
If possible, row reduce A to E using elementary row
operations on the entire matrix [A : E]. The result will be
the matrix [E : A
-1
]. If this is not possible, then A is not
invertible.
1
0
0
0
1
0
0
0
1
3
2
6
1
0
1
0
1
1
)
6
(
1
)
1
(
1
r
r
1
0
6
0
1
1
0
0
1
3
4
0
1
1
0
0
1
1
~
)
4
(
2
)
1
(
2
r
r
Algorithm (
Gauss-Jordan Elimination
)
1
4
2
1
3
1
1
3
2
1
0
0
0
1
0
0
0
1
~
1
4
2
1
3
3
1
3
2
1
A
•
Check your work by multiplying to see A
-1
A = A A
-1
= E
1
0
0
0
1
0
0
0
1
1
4
2
1
3
3
1
3
2
3
2
6
1
0
1
0
1
1
1
4
2
0
1
1
0
1
0
1
0
0
1
1
0
1
0
1
~
)
1
(
1
4
2
0
1
1
0
1
0
1
0
0
1
1
0
1
0
1
~
)
1
(
3
)
1
(
3
r
r
1
0
0
0
1
0
0
0
1
3
2
6
1
0
1
0
1
1
1
4
2
1
3
3
1
3
2
Example
:
Find the inverse of the following matrix
2
3
2
2
1
3
0
2
1
A
1
0
0
0
1
0
0
0
1
2
3
2
2
1
3
0
2
1
E
A
)
6
(
1
)
1
(
1
r
r
1
0
2
0
1
3
0
0
1
2
7
0
2
7
0
0
2
1
~
)
1
(
2
r
1
1
1
0
1
3
0
0
1
0
0
0
2
7
0
0
2
1
~
„A portion” of the
matrix has a row of
zeros
It is not posible to
rewrite the matrix [A :
E] to [E : A
-1
]
A has no
inverse
Finding the Inverse of Matrix by Its Adjoint
Definition 2.5:
Definition 2.5:
The
The
adjoint
adjoint
of matrix A is denoted
of matrix A is denoted
adj
adj
(A) and has
(A) and has
folowing form
folowing form
Where
Where
is a
is a
cofactor
cofactor
, and
, and
is a minor of the element .
is a minor of the element .
T
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
adj
nn
n
n
n
n
n
n
3
2
1
3
33
32
31
2
23
22
21
1
13
12
11
A
ij
j
i
ij
C
A
det
1
ij
A
det
ij
a
Find the
adjoint of
,
2
0
1
1
2
0
2
3
1
A
Example
:
,
2
0
1
1
2
0
2
3
1
A
4
2
0
1
2
det
1
1
1
11
C
Continuing this process produces the following matrix of cofactors of A
2
1
7
3
0
6
2
1
4
2
0
3
1
1
0
2
1
1
2
2
3
0
1
3
1
2
1
2
1
2
0
2
3
0
1
2
0
2
1
1
0
2
0
1
2
2
3
2
1
0
1
7
6
4
)
(A
adj
Matrix of cofactors
Theorem 2.5:
Theorem 2.5:
If A is an n n invertible matrix, then
)
(
)
det(
1
A
A
A
1
adj
Example
:
Find the inverse of the following matrix
2
3
2
2
1
3
0
2
1
A
Proof: without proof
2
3
2
2
1
3
0
2
1
det
)
det(A
0
12
6
8
2
matrix A is
singular !!!
A
-1
does not exist
3
2
6
1
0
1
0
1
1
A
Example
:
1
3
2
6
3
2
6
1
0
1
0
1
1
det
Find the inverse of matrix
A is invertible
3
2
6
1
0
1
0
1
1
A
2
3
2
1
0
)
1
(
1
1
11
C
T
adj
2
)
(A
3
2
6
1
0
1
0
1
1
A
Example
:
1
3
2
6
3
2
6
1
0
1
0
1
1
det
Find the inverse of matrix
A is invertible
3
2
6
1
0
1
0
1
1
A
3
3
6
1
1
)
1
(
2
1
12
C
T
adj
2
)
(A
T
adj
3
2
)
(A
3
2
6
1
0
1
0
1
1
A
Example
:
1
3
2
6
3
2
6
1
0
1
0
1
1
det
Find the inverse of matrix
A is invertible
3
2
6
1
0
1
0
1
1
A
2
2
6
0
1
)
1
(
3
1
13
C
T
adj
3
2
)
(A
T
adj
2
3
2
)
(A
3
2
6
1
0
1
0
1
1
A
Example
:
1
3
2
6
3
2
6
1
0
1
0
1
1
det
Find the inverse of matrix
A is invertible
3
2
6
1
0
1
0
1
1
A
3
3
2
0
1
)
1
(
1
2
21
C
T
adj
2
3
2
)
(A
T
adj
3
2
3
2
)
(A
3
2
6
1
0
1
0
1
1
A
Example
:
1
3
2
6
3
2
6
1
0
1
0
1
1
det
Find the inverse of matrix
A is invertible
3
2
6
1
0
1
0
1
1
A
3
3
6
0
1
)
1
(
2
2
22
C
T
adj
3
2
3
2
)
(A
T
adj
3
3
2
3
2
)
(A
T
adj
3
3
2
3
2
)
(A
3
2
6
1
0
1
0
1
1
A
Example
:
1
3
2
6
3
2
6
1
0
1
0
1
1
det
Find the inverse of matrix
A is invertible
3
2
6
1
0
1
0
1
1
A
23
C
4
2
6
1
1
)
1
(
3
2
T
adj
4
3
3
2
3
2
)
(A
T
adj
4
3
3
2
3
2
)
(A
3
2
6
1
0
1
0
1
1
A
Example
:
1
3
2
6
3
2
6
1
0
1
0
1
1
det
Find the inverse of matrix
A is invertible
3
2
6
1
0
1
0
1
1
A
31
C
1
1
0
0
1
)
1
(
1
3
T
adj
1
4
3
3
2
3
2
)
(A
T
adj
1
4
3
3
2
3
2
)
(A
3
2
6
1
0
1
0
1
1
A
Example
:
1
3
2
6
3
2
6
1
0
1
0
1
1
det
Find the inverse of matrix
A is invertible
3
2
6
1
0
1
0
1
1
A
1
1
1
0
1
)
1
(
2
3
32
C
T
adj
1
1
4
3
3
2
3
2
)
(A
T
adj
1
1
4
3
3
2
3
2
)
(A
3
2
6
1
0
1
0
1
1
A
Example
:
1
3
2
6
3
2
6
1
0
1
0
1
1
det
Find the inverse of matrix
A is invertible
3
2
6
1
0
1
0
1
1
A
1
0
1
1
1
)
1
(
3
3
33
C
T
adj
1
1
1
4
3
3
2
3
2
)
(A
1
4
2
1
3
3
1
3
2
1
4
2
1
3
3
1
3
2
1
1
1
A