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MATRIX ALGEBRA
Chapter 1
CHAPTER 1 (10 HOURS=5 CLASSES)
1.1 Defini/on of a matrix
1.2 Iden/ty Matrix, Diagonal Matrix, Symmetric Matrix
1.3 Sums and Scalar Mul/ples
1.4 Matrix Mul/plica/on
1.5 Proper/es of Matrix Mul/plica/on
1.6 Transpose of a Matrix
Addi/on sec/ons
2.5 Rank of matrix
2.6 Inverse matrix
EXAMPLES
An 2x3 matrix
// 2 rows, 3 columns
Read: two by three matrix
Read: row by row
Seven minus three 1 half / three minus 5 zero
7 -3 1/2
3 -5 0
(1,3)-entry
a[1,3] = 1/2
a
13
= ½
Read: a_one_three entry
A =
MATRIX
CLASSIFICATION
1. By shape: rectangular; square; row; column
2. By value of entries: zero; identity; scalar
3. By characteristics: diagonal; triangular; symmetric
1.2 SUM AND SCALAR MULTIPLES
Equality
AddiJon
SubtracJon
Scalar MulJplicaJon (scalar mulJples)
EQUALITY
Ex. Consider 4 matrices:
A
=
(
1
2
3
4
)
; B
=
(
1
3
)
;C
=
(
1
3
)
; and D
=
(
1
2
x
4
)
Discuss the possibility of equality between these matrices.
QUICK CHECK
Given
discuss the possibility that A = B, B = C, A = C
ADDITION AND SUBTRACTION
Addi/on A + B = [a
ij
+ b
ij
]
Subtrac/on A – B = [a
ij
– b
ij
]
Example:
A)
B)
The same size
matrices
SCALAR MULTIPLICATION
A
=
(
1
2
−3
5
)
2 A
=
2×
(
1
2
−3
5
)
=
(
2
4
−6
10
)
− A
=
(
−1
)
A
=
(
−1
−2
3
−5
)
B
=
(
12
4
8
0
10
2
)
=
2×
(
6
2
4
0
5
1
)
1.4 MATRIX MULTIPLICATION
A
m
×
n
. B
n
×
p
= C
m
×
p
//suitable size
The entry c
ij
= (row i of A).(column j of B)
2
⎟
⎟
⎟
⎠
⎞
⎜
⎜
⎜
⎝
⎛
=
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
−
×
⎟
⎟
⎟
⎠
⎞
⎜
⎜
⎜
⎝
⎛
−
−
0
1
2
1
2
1
0
1
0
2
1
0
2
1
1.1+2.1
3
4
1
2
-1
-2
-1
0
-2
0
2
-4
-1
-2
-1
0
-2
0
2
1
4
3
APPLICATION
peanuts
soda
hot dogs
set A
8
5
12
set B
15
7
13
selling price
store 1
store 2
store 3
store 4
peanuts
2
2.5
2
2.5
soda
2.5
2
2.75
2
hot dogs
3
3
2.5
3
2 gift sets: A&B.
4 stores
different prices
Find the best choice?
Matrix
Matrix
A SIMPLE APPLICATION
peanuts
soda
hot dogs
set A
8
5
12
set B
15
7
13
selling price
store 1
store 2
store 3
store 4
peanuts
2
2.5
2
2.5
soda
2.5
2
2.75
2
hot dogs
3
3
2.5
3
B
=
[
2
2.5
2
2.5
2.5
2
2.75
2
3
3
2.5
3
]
dimension : item × store
A
=
[
8
5
12
15
7
13
]
row × col
=
set × item
APPLICATION
store 1
store 2
store 3
store 4
set A
64.5
66
59.75
66
set B
86.5
87.5
81.75
90.5
Quantity
A
=
[
8
5
12
15
7
13
]
row × col
=
set × item
Price
B
=
[
2
2.5
2
2.5
2.5
2
2.75
2
3
3
2.5
3
]
dimension : item × store
Quantity × Price
=
A × B
=
[
64.5
?
?
?
?
?
?
?
]
Dimension :? ×?
QUICK CHECK
Perform the calculation:
[
6
3
1
1
4
−2
4
−1
5
]
×
(
x
1
x
2
x
3
)
=
0
WHY NO DIVISION
TERMS
In view of the general rule AB not equal BA, the terms premul/ply
and postmul/ply are o_en used to specify the order of
mul/plica/on.
In the product A.B,
the matrix B is said to be premul/plied by A,
and A to be postmul/plied by B.
Or in brief le_ and right mul/plied
07/1/2025 W2.1
Chapter 1
2.5 RANK OF THE MATRIX
Row echelon matrix
Reduced row echelon matrix
Rank of a matrix
EXAMPLE R.E.F
B
=
[
1
−2
3
5
0
10
3
6
0
0
1
−3
]
zero rows
leading entry right
Below leading entry are 0
A
=
[
1
22
3
0
0
1
0
1
−3
]
C
=
[
0
0
0
0
1
−2
3
5
0
1
3
6
0
0
1
−3
]
D
=
[
1
−2
3
5
0
1
3
6
0
0
1
−3
0
0
0
0
]
E
=
[
10
−2
3
5
0
3
3
6
0
0
6
−3
]
F
=
[
1
−2
3
5
0
−1
3
6
0
0
1
−3
]
EXAMPLE R.R.E.F
B
=
[
1
−2
3
5
0
10
3
6
0
0
1
−3
]
zero rows
Leading 1, right
Below and above leading entry are 0
A
=
[
1
22
3
0
0
1
0
1
−3
]
C
=
[
0
0
0
0
1
−2
3
5
0
1
3
6
0
0
1
−3
]
D
=
[
1
0
0
5
0
1
0
6
0
0
1
−3
0
0
0
0
]
E
=
[
1
0
3
5
0
3
3
6
0
0
6
12
]
F
=
[
1
0
0
5
0
1
0
6
0
0
1
−3
0
0
0
0
]
EXAMPLE
Let the matrix A given as
Complete the diagram below
A
=
[
0
2
3
2
−3
5
−1
2
0
]
r
3
←→r
1
A
A1
r
2
→r
2
+
2 r
1
r
3
→r
3
− 2 r
2
A1
A3
A2
EXAMPLE
Carry the matrix
A) to row-echelon matrix
B) to reduced row-echelon matrix
2
6
2
2
2
3
11
4
3
11
3
0
−
⎡
⎤
⎢
⎥
−
−
⎢
⎥
⎢
⎥
⎣
⎦
2
2
3
3
3
2
3
1
1
2
3
7
1
3
1
1
1
3
1
1
1
3
1
1
0
1
3
2
0
1
3
2
0
1
3
2
0
2
6
3
0
0
0
7
0
0
0
1
r
r
r
r
r
r
r
→
→−
→−
+
−
−
−
⎡
⎤
⎡
⎤
⎡
⎤
⎢
⎥
⎢
⎥
⎢
⎥
→
→
→
⎢
⎥
⎢
⎥
⎢
⎥
⎢
⎥
⎢
⎥
⎢
⎥
−
−
⎣
⎦
⎣
⎦
⎣
⎦
2
2
1
1
1
3
3
1
1
2
3
2
2
6
2
2
1
3
1
1
1
3
1
1
2
3
11
4
2
3
11
4
0
3
9
6
3
11
3
0
3
11
3
0
0
2
6
3
r
r
r
r
r
r
r
r
→ +
→
→ −
−
−
−
⎡
⎤
⎡
⎤
⎡
⎤
⎢
⎥
⎢
⎥
⎢
⎥
−
−
→ −
−
→
⎢
⎥
⎢
⎥
⎢
⎥
⎢
⎥
⎢
⎥
⎢
⎥
−
⎣
⎦
⎣
⎦
⎣
⎦
row-echelon
matrix
But not
reduced yet
EXAMPLE
reduced row-echelon matrix
2
2
3
1
1
3
1
2
1
2
3
1
3
1
1
1
3
1
0
1
0
10
0
0
1
3
2
0
1
3
0
0
1
3
0
0
0
0
1
0
0
0
1
0
0
0
1
r
r
r
r
r
r
r
r
r
→ −
→ −
→−
+
−
−
−
⎡
⎤
⎡
⎤
⎡
⎤
⎢
⎥
⎢
⎥
⎢
⎥
→
→
⎢
⎥
⎢
⎥
⎢
⎥
⎢
⎥
⎢
⎥
⎢
⎥
⎣
⎦
⎣
⎦
⎣
⎦
EXAMPLE
THE RANK OF A MATRIX
1
0
*
0
0
1
*
0
0
0
0
0
⎡
⎤
⎢
⎥
⎢
⎥
⎢
⎥
⎣
⎦
The rank of matrix
A
is the number of leading entry in any
row-echelon matrix to which
A
can be carried by row
opera/ons.
The rank of the matrix A, rank(A), is the
number of leading ones
in the reduced row-echelon form of A.
rank(A)=2
r(A)=2
The rank of matrix
A
is the number of non-zero rows in any
row-echelon matrix to which
A
can be carried by row
opera/ons.
EXAMPLE
EXAMPLE
Determine the rank of the matrix
EXAMPLE
Determine all value(s) m such that r(A)=3 where
Conclusion ???
PROPERTIES
10/1/2025 W2.2
Chapter 1
Review
1. Matrix Mul/plica/on
2. Transposi/on
3. Row reduc/ons
4. REF; RREF and rank.
5. Inverse matrix
Inverse matrix
Q1. What?
Q2.0 When?
Q2. How?
Q3. Why? Where?
Q3. WHY?
A . X
=
B→ X
=
?
Solving Linear System
Undoing Transformations
...
EXAMPLE
Let the matrix A given as
Complete the diagram below
A
=
[
0
2
3
2
−3
5
−1
2
0
]
r
3
←→r
1
A
A1
r
2
→r
2
+
2 r
1
r
3
→r
3
− 2 r
2
A1
A3
A2
REMARK
Not all square matrices are inver/ble.
There are many nonzero matrices that are not inver/ble
An inver/ble matrix is said to be nonsingular.
A square matrix with no inverse is called a singular
matrix.
EXAMPLE
PROPERTIES
(CancellaJon Laws) Let A be an inver/ble matrix.
If A and B are inver/ble n ×n matrices
THEOREM 2.4.4
EXISTENCE OF INVERSE MATRIX
For an nxn matrix A, the following statements below are
equivalent.
EXAMPLE
A
-1
[A | I
n
]
→
[I
n
|A
-1
]