This document provides an overview of matrices and matrix operations. It begins by stating the objectives of understanding matrix characteristics, applying basic matrix operations, knowing inverse matrices up to 3x3, and solving simultaneous linear equations up to 3 variables. It then defines what a matrix is, discusses matrix dimensions and types of matrices. The document outlines various matrix operations including addition, subtraction, multiplication and scalar multiplication. It provides examples of how to perform these operations. It also covers the transpose of a matrix and inverse matrices.

1. BA301
ENGINEERING
MATHEMATICS 3

3. INDEX
OBJECTIVES
DIMENSIONS
TYPES OF MATRICES
OPERATION ON MATRICES
- Addition
- Subtraction
- Multiplication
- Scalar Multiplication
SOLVING MATRICES PROBLEMS
TRANSPOSE OF MATRIX
WHAT IS A MATRIX
TUTORIAL A
SOLVING SIMULTANEOUS EQUATION
INVERSE MATRIX

4. OBJECTIVES:
- Know characteristics of matrices
- Apply basic operations on matrices
- Know the inverse matrices (up to
3X3)
- Solve simultaneous linear equations
up to 3 variables

5. OBJECTIVES :
-Know
characteristics
of matrices
-Apply basic
operations on
matrices
-Know the
inverse
matrices (up to
3X3)
-Solve
simultaneous
linear
equations up to
3 variables
Matrix : A rectangular arrangement of
numbers in rows and columns
Order/Dimensions/Size of a
matrix is the number of the rows
and columns
Entries /Elements are the numbers
in the matrix

6. OBJECTIVES :
-Know
characteristics
of matrices
-Apply basic
operations on
matrices
-Know the
inverse
matrices (up to
3X3)
-Solve
simultaneous
linear
equations up to
3 variables
row






2
3
4
0
1
2
column
(# no. of row) x ( #
no. of column)
DIMENSIONS OF THE MATRIX :
So……..
This is a 2 X 3 matrix

7. OBJECTIVES :
-Know
characteristics
of matrices
-Apply basic
operations on
matrices
-Know the
inverse
matrices (up to
3X3)
-Solve
simultaneous
linear
equations up to
3 variables
A matrix with m rows and n columns is
called a matrix with dimensions m x n
Example : Find the dimensions of the
matrix.






1
9
7
3
2
4
)
a










2
9
3
7
0
5
)
b
 
3
8
1
) 
c










5
1
2
3
4
8
10
9
7
)
d
2 X 3 3 X 2
1 X 3 3 X 3

8. OBJECTIVES :
-Know
characteristics
of matrices
-Apply basic
operations on
matrices
-Know the
inverse
matrices (up to
3X3)
-Solve
simultaneous
linear
equations up to
3 variables
Find the dimensions of the following matrix
 
5
)
a






7
1
)
b






2
3
4
9
)
c
1 X 1
2 X 1
2 X 2

9. OBJECTIVES :
-Know
characteristics
of matrices
-Apply basic
operations on
matrices
-Know the
inverse
matrices (up to
3X3)
-Solve
simultaneous
linear
equations up to
3 variables

10. No of rows = No of column







2
3
4
5
T 2 X 2
Square
matrix











5
3
4
0
2
6
1
4
8
A 3 X 3
Square
matrix
OBJECTIVES :
-Know
characteristics
of matrices
-Apply basic
operations on
matrices
-Know the
inverse
matrices (up to
3X3)
-Solve
simultaneous
linear
equations up to
3 variables

11. All entries = 0 EXCEPT at least one
Of the diagonal entries











1
0
0
0
5
0
0
0
3
T
diago
nal











0
0
1
0
5
0
3
0
0
T
Diagon
al
Matrix
OBJECTIVES :
-Know
characteristics
of matrices
-Apply basic
operations on
matrices
-Know the
inverse
matrices (up to
3X3)
-Solve
simultaneous
linear
equations up to
3 variables

12. Diagonal entries = 1
All other entries = 0











1
0
0
0
1
0
0
0
1
I











0
0
1
0
1
0
1
0
0
I







1
0
0
1
I







0
1
1
0
I
exampl
e
OBJECTIVES :
-Know
characteristics
of matrices
-Apply basic
operations on
matrices
-Know the
inverse
matrices (up to
3X3)
-Solve
simultaneous
linear
equations up to
3 variables

13. Element in row change into column
and element in column change into rows











1
8
1
6
5
7
4
2
3
B











1
6
4
8
5
2
1
7
3
T
B
OBJECTIVES :
-Know
characteristics
of matrices
-Apply basic
operations on
matrices
-Know the
inverse
matrices (up to
3X3)
-Solve
simultaneous
linear
equations up to
3 variables

14. OBJECTIVES :
-Know
characteristics
of matrices
-Apply basic
operations on
matrices
-Know the
inverse
matrices (up to
3X3)
-Solve
simultaneous
linear
equations up to
3 variables

15. OBJECTIVES :
-Know
characteristics
of matrices
-Apply basic
operations on
matrices
-Know the
inverse
matrices (up to
3X3)
-Solve
simultaneous
linear
equations up to
3 variables

16. -Both matrix must have the same
dimensions to do addition
operation
-To add matrices, we add the
corresponding elements
OBJECTIVES :
-Know
characteristics
of matrices
-Apply basic
operations on
matrices
-Know the
inverse
matrices (up to
3X3)
-Solve
simultaneous
linear
equations up to
3 variables

17. Example :







4
5
1
2
A 






8
7
6
5
B
Find A + B







 B
A 2 + 5 1 + 6
5 + 7 4+ 8








12
12
7
7
B
A
OBJECTIVES :
-Know
characteristics
of matrices
-Apply basic
operations on
matrices
-Know the
inverse
matrices (up to
3X3)
-Solve
simultaneous
linear
equations up to
3 variables
CALC
GEO

18. Find A + B
   
4
5
3
2
) 


 B
and
A
a




















3
2
3
9
4
2
9
8
7
2
3
8
) B
and
A
b
 
7
7 

 B
A











12
10
4
11
7
6
B
A
OBJECTIVES :
-Know
characteristics
of matrices
-Apply basic
operations on
matrices
-Know the
inverse
matrices (up to
3X3)
-Solve
simultaneous
linear
equations up to
3 variables

19. -Both matrix must have the same
dimensions to do subtraction operation
-To subtract matrices, we subtract the
corresponding elements
OBJECTIVES :
-Know
characteristics
of matrices
-Apply basic
operations on
matrices
-Know the
inverse
matrices (up to
3X3)
-Solve
simultaneous
linear
equations up to
3 variables

20. Example :







4
5
1
2
A 






8
7
6
5
B
Find A - B







 B
A 2 -
5
1 -
6
5 -
7
4- 8












4
2
5
3
B
A
OBJECTIVES :
-Know
characteristics
of matrices
-Apply basic
operations on
matrices
-Know the
inverse
matrices (up to
3X3)
-Solve
simultaneous
linear
equations up to
3 variables
GEO

21. Find A - B
   
4
5
3
2
) 


 B
and
A
a




















3
2
3
9
4
2
9
8
7
2
3
8
) B
and
A
b
 
1
3


 B
A











6
6
10
7
1
10
B
A
OBJECTIVES :
-Know
characteristics
of matrices
-Apply basic
operations on
matrices
-Know the
inverse
matrices (up to
3X3)
-Solve
simultaneous
linear
equations up to
3 variables

22. 









2
9
7
6
1
3
4
5
2










k
k
k
k
k
k
k
k
k
2
9
7
6
1
3
4
5
2
Multiply each element inside the matrix
by
OBJECTIVES :
-Know
characteristics
of matrices
-Apply basic
operations on
matrices
-Know the
inverse
matrices (up to
3X3)
-Solve
simultaneous
linear
equations up to
3 variables

23. Example :








6
3
4
2
5
Solve







5 X
2
5 X -
4
5 X -
3
5 X 6









30
15
20
10
OBJECTIVES :
-Know
characteristics
of matrices
-Apply basic
operations on
matrices
-Know the
inverse
matrices (up to
3X3)
-Solve
simultaneous
linear
equations up to
3 variables
GEO

24. Solve the following :
 
12
9
3
4
) 
a
 
16
12 











9
8
7
2
3
8
5
)
b










45
40
35
10
15
40
OBJECTIVES :
-Know
characteristics
of matrices
-Apply basic
operations on
matrices
-Know the
inverse
matrices (up to
3X3)
-Solve
simultaneous
linear
equations up to
3 variables

25. Look at the dimension of matrices A and B to do
MULTIPLICATIONS
must be same
size of product
If the numbers of columns of A does
NOT EQUAL the number of rows of B,
then…
A X B is
UNDEFINED..!!
OBJECTIVES :
-Know
characteristics
of matrices
-Apply basic
operations on
matrices
-Know the
inverse
matrices (up to
3X3)
-Solve
simultaneous
linear
equations up to
3 variables

26. Example :







7
4
3
1
5
2
A











8
3
2
B
and
Find A X B.


















8
3
2
7
4
3
1
5
2
B
X
A
2 X 3 3 X 1
sam
e
Size of product =
2 X 1
VERY
IMPORTANT!!
OBJECTIVES :
-Know
characteristics
of matrices
-Apply basic
operations on
matrices
-Know the
inverse
matrices (up to
3X3)
-Solve
simultaneous
linear
equations up to
3 variables

27. 






AXB (2 x 2) + (5x 3) + (1x 8)
(3x2) + (4x3) + (7x8)











56
12
6
8
15
4
AXB







74
27
AXB


















8
3
2
7
4
3
1
5
2
B
X
A
OBJECTIVES :
-Know
characteristics
of matrices
-Apply basic
operations on
matrices
-Know the
inverse
matrices (up to
3X3)
-Solve
simultaneous
linear
equations up to
3 variables
GEO

28. .
.
12
6
4
5
7
8
2
5
1
2
3
9
) AXB
Find
B
and
A
a 




 














  .
.
7
5
6
4
5
8
) PXQ
Find
Q
and
P
b























22
8
20
9
81
60
AXB
 
5


PXQ
OBJECTIVES :
-Know
characteristics
of matrices
-Apply basic
operations on
matrices
-Know the
inverse
matrices (up to
3X3)
-Solve
simultaneous
linear
equations up to
3 variables

29. .
.
20
2
5
7
5
3
9
6
8
3
5
4
2
3
1
8
6
7
) MXN
Find
N
and
M
c









 



























61
43
32
70
13
27
181
56
78
MXN













 

2
8
5
3
1
4
) U
and
T
d







14
34
MXN
OBJECTIVES :
-Know
characteristics
of matrices
-Apply basic
operations on
matrices
-Know the
inverse
matrices (up to
3X3)
-Solve
simultaneous
linear
equations up to
3 variables

30. Example :
y
and
x
Find
x
y
x
.
1
30
12
19
10
6
6
8
1
2
8
2
5
3
3 





















3





 9x-15 6
24 6y+3














1
30
12
19
10
6
6
8x








13
6
30
12
15
17
y
x







1
30
12
19
OBJECTIVES :
-Know
characteristics
of matrices
-Apply basic
operations on
matrices
-Know the
inverse
matrices (up to
3X3)
-Solve
simultaneous
linear
equations up to
3 variables

31. 17 x - 15
=19
17 x = 15 +
19
17 x =
34
x = 34/17
x = 2
6 y + 13 =
1
6 y = 13 -
1
6y = 12
y =
12/6
y =
2
OBJECTIVES :
-Know
characteristics
of matrices
-Apply basic
operations on
matrices
-Know the
inverse
matrices (up to
3X3)
-Solve
simultaneous
linear
equations up to
3 variables

32. Tips
:
colum
n
row
OBJECTIVES :
-Know
characteristics
of matrices
-Apply basic
operations on
matrices
-Know the
inverse
matrices (up to
3X3)
-Solve
simultaneous
linear
equations up to
3 variables

33. Example :







T
A
.
.
2
5
1
2
3
9
T
A
Find
A
Given













9 2 5
-3 1 -
2
OBJECTIVES :
-Know
characteristics
of matrices
-Apply basic
operations on
matrices
-Know the
inverse
matrices (up to
3X3)
-Solve
simultaneous
linear
equations up to
3 variables
GEO

34. 




 

12
6
4
5
7
8
) B
a
Find transpose of the following
matrices :












12
5
6
7
4
8
T
B











6
7
8
) P
b
 
6
7
8

T
P
OBJECTIVES :
-Know
characteristics
of matrices
-Apply basic
operations on
matrices
-Know the
inverse
matrices (up to
3X3)
-Solve
simultaneous
linear
equations up to
3 variables

35. C
B
d
A
B
D
c
C
A
b
B
A
a
Find
D
C
B
A



































)
)
(
)
)
)
:
.
8
4
5
3
5
3
6
2
4
3
5
9
8
3
.
1
BA
b
AB
a
Find
Find
B
and
A
If
)
)
:
:
.
6
2
4
0
5
9
6
1
.
2 















.
.
3
5
2
6
3
4
0
5
9
7
6
2
.
3 MN
Find
N
and
M
If






















Answe
r
Answe
r
Answe
r
OBJECTIVES :
-Know
characteristics
of matrices
-Apply basic
operations on
matrices
-Know the
inverse
matrices (up to
3X3)
-Solve
simultaneous
linear
equations up to
3 variables

36. 2
)
)
:
.
1
3
9
3
5
7
7
6
1
4
5
5
9
6
1
.
5
B
b
BA
a
Find
B
and
A
If






















T
T
B
A
b
B
A
a
Find
B
and
A
Given
)
(
)
)
:
.
2
4
8
8
6
9
3
4
5
5
6
2
.
6




















B
A
b
B
A
a
Find
B
and
A
matrix
Given
























)
)
:
.
3
9
4
8
6
1
3
0
5
9
6
2
.
4
Answe
r
Answe
r
Answe
r
OBJECTIVES :
-Know
characteristics
of matrices
-Apply basic
operations on
matrices
-Know the
inverse
matrices (up to
3X3)
-Solve
simultaneous
linear
equations up to
3 variables

37. solution
no
C
B
d
A
B
D
c
solution
no
C
A
b
B
A
a
Find
D
C
B
A






















































)
3
7
1
3
)
(
)
)
1
7
12
6
)
:
.
8
4
5
3
5
3
6
2
4
3
5
9
8
3
.
1
TUTORIAL A
OBJECTIVES :
-Know
characteristics
of matrices
-Apply basic
operations on
matrices
-Know the
inverse
matrices (up to
3X3)
-Solve
simultaneous
linear
equations up to
3 variables

38. 


































42
56
20
36
)
6
10
32
12
)
:
:
.
6
2
4
0
5
9
6
1
.
2
BA
b
AB
a
Find
Find
B
and
A
If
TUTORIAL A
OBJECTIVES :
-Know
characteristics
of matrices
-Apply basic
operations on
matrices
-Know the
inverse
matrices (up to
3X3)
-Solve
simultaneous
linear
equations up to
3 variables

39. 











































41
43
55
.
.
8
3
9
2
4
8
7
6
1
3
5
2
6
3
4
0
5
9
7
6
2
.
3
MN
MN
Find
P
and
N
M
If
TUTORIAL A
OBJECTIVES :
-Know
characteristics
of matrices
-Apply basic
operations on
matrices
-Know the
inverse
matrices (up to
3X3)
-Solve
simultaneous
linear
equations up to
3 variables

40. 














































0
9
1
1
0
1
)
6
9
9
17
12
3
)
:
.
3
9
4
8
6
1
3
0
5
9
6
2
.
4
B
A
b
B
A
a
Find
B
and
A
matrix
Given
TUTORIAL A
OBJECTIVES :
-Know
characteristics
of matrices
-Apply basic
operations on
matrices
-Know the
inverse
matrices (up to
3X3)
-Solve
simultaneous
linear
equations up to
3 variables

41. 











































73
72
39
67
76
69
32
57
106
)
73
41
79
67
64
90
)
:
.
1
3
9
3
5
7
7
6
1
4
5
5
9
6
1
.
5
2
B
b
BA
a
Find
B
and
A
If
TUTORIAL A
OBJECTIVES :
-Know
characteristics
of matrices
-Apply basic
operations on
matrices
-Know the
inverse
matrices (up to
3X3)
-Solve
simultaneous
linear
equations up to
3 variables

42. 





































5
9
14
12
11
11
)
5
12
9
11
14
11
)
:
.
2
4
8
8
6
9
3
4
5
5
6
2
.
6
B
A
b
B
A
a
Find
B
and
A
Given
T
T
TUTORIAL A
OBJECTIVES :
-Know
characteristics
of matrices
-Apply basic
operations on
matrices
-Know the
inverse
matrices (up to
3X3)
-Solve
simultaneous
linear
equations up to
3 variables

43. 2 X 2
3 X 3
Formula :
adjA
A
A 

 1
1
.
.
6
5
4
5 1







 A
Find
A
)
4
(
5
)
6
(
5 

A
20
30

A
10

A
Example :
1. Determinant A
GEO
2 X 2
OBJECTIVES :
-Know
characteristics
of matrices
-Apply basic
operations on
matrices
-Know the
inverse
matrices (up to
3X3)
-Solve
simultaneous
linear
equations up to
3 variables

44. 2. Minor A







6
5
4
5
A







6
5
4
5
M What you can
see??
6
11 
m 5
12 
m
4
21 
m 5
22 
m







5
4
5
6
Minor
OBJECTIVES :
-Know
characteristics
of matrices
-Apply basic
operations on
matrices
-Know the
inverse
matrices (up to
3X3)
-Solve
simultaneous
linear
equations up to
3 variables

45. 3. Cofactor A







5
4
5
6
Minor









5
4
5
6
Cofactor
4. Adjoint A









5
5
4
6
A
Adj
adjA
A
A 

 1
1











5
5
4
6
10
1
1
A
5. Inverse A
















2
1
2
1
5
2
5
3
10
1
GEO
OBJECTIVES :
-Know
characteristics
of matrices
-Apply basic
operations on
matrices
-Know the
inverse
matrices (up to
3X3)
-Solve
simultaneous
linear
equations up to
3 variables

46. Formula :
Example :
.
sin
.
3
1
7
2
3
5
1
1
2
1
Matrix
Inverse
g
u
by
A
Find
A 












1. Determinant A













7
7
3
1
7
3
5
2
3
5
1
2
1
1
2
A
 
3
3
2 



A  
7
2
1 
  
7
5
1 
  
1
3
7 

  
2
2
1 
  
1
5
3 

 
35
14
18 



A  
15
4
21 


33

A
3 X 3
adjA
A
A 

 1
1
OBJECTIVES :
-Know
characteristics
of matrices
-Apply basic
operations on
matrices
-Know the
inverse
matrices (up to
3X3)
-Solve
simultaneous
linear
equations up to
3 variables

47. 2. Minor












3
1
7
2
3
5
1
1
2
A
   
11
2
1
3
3
3
1
2
3
11
11
11












m
m
m
   
1
2
7
3
5
3
7
2
5
12
12
12










m
m
m
   
26
3
7
1
5
1
7
3
5
13
13
13









 

m
m
m
   
2
1
1
3
1
3
1
1
1
21
21
21










m
m
m
   
1
7
1
3
2
3
7
1
2
22
22
22











m
m
m
   
5
1
7
1
2
1
7
1
2
23
23
23











m
m
m
   
5
3
1
2
1
2
3
1
1
31
31
31












m
m
m
   
1
5
1
2
2
2
5
1
2
32
32
32











m
m
m
   
11
33
5
1
3
2
33
3
5
1
2
33













m
m
m
OBJECTIVES :
-Know
characteristics
of matrices
-Apply basic
operations on
matrices
-Know the
inverse
matrices (up to
3X3)
-Solve
simultaneous
linear
equations up to
3 variables

48. 
















11
1
5
5
1
2
26
1
11
M
3. Cofactor
















11
1
5
5
1
2
26
1
11
K
















11
1
5
5
1
2
26
1
11
K
4. Adjoint
















11
5
26
1
1
1
5
2
11
A
adj
OBJECTIVES :
-Know
characteristics
of matrices
-Apply basic
operations on
matrices
-Know the
inverse
matrices (up to
3X3)
-Solve
simultaneous
linear
equations up to
3 variables

49. 5. Inverse A

















11
5
26
1
1
1
5
2
11
33
1
1
A

























3
1
33
5
33
26
33
1
33
1
33
1
33
5
33
2
3
1
33
1
1
A
OBJECTIVES :
-Know
characteristics
of matrices
-Apply basic
operations on
matrices
-Know the
inverse
matrices (up to
3X3)
-Solve
simultaneous
linear
equations up to
3 variables

50. Cramer’s Rule Formula :
A
A
z
A
A
y
A
A
x z
y
x



INVERSE MATRIX CRAMER’S RULE
Inverse Matrix Formula :
adjA
A
A 

 1
1
OBJECTIVES :
-Know
characteristics
of matrices
-Apply basic
operations on
matrices
-Know the
inverse
matrices (up to
3X3)
-Solve
simultaneous
linear
equations up to
3 variables

51. CRAMER’S RULE
Example :
.
'
sin
,
8
4
8
2
5
9
2
6
4
7
5
Rule
s
Cramer
g
u
by
z
and
y
x
Find
z
y
x
z
y
x
z
y
x

























8
2
4
8
2
2
6
9
2
6
1
5
7
1
5
A
1. Determinant A

A 4
2
5 


 2
9
1 

 8
6
7 
 7
2
2 

 5
9
8 
 1
6
4 



    ]
24
360
28
[
336
18
40 






A
2

A
OBJECTIVES :
-Know
characteristics
of matrices
-Apply basic
operations on
matrices
-Know the
inverse
matrices (up to
3X3)
-Solve
simultaneous
linear
equations up to
3 variables

52. 2. Determinant Ax
















8
8
4
8
2
5
9
2
1
4
7
1
x
A
4
5
8
5
6
2

x
A 4
2
4 


 8
9
1 

 8
5
7 
 7
2
8 

 4
9
8 
 1
5
4 



    ]
20
288
112
[
280
72
32 






x
A
44

x
A
3. Determinant Ay












8
2
4
2
5
6
9
6
4
5
7
5
x
A
4
5
8
-1
-2
8

y
A 4
5
5 

 2
9
4 
 8
6
7 
 7
5
2 
 5
9
8 
 4
6
4 


  ]
96
360
70
[
336
72
100 






y
A
26


y
A
OBJECTIVES :
-Know characteristics of matrices
-Apply basic operations on matrices
-Know the inverse matrices (up to 3X3)
-Solve simultaneous linear equations up to
3 variables

53. 4. Determinant Az















8
2
8
8
2
2
6
5
2
6
1
5
4
1
5
z
A

z
A 8
2
5 

 8
6
4 

2
5
1 

 4
2
2 

 5
5
8 
 1
6
8 


  ]
48
200
16
[
192
10
80 







z
A
34


z
A
22
2
44


x 13
2
26




y 17
2
34




z
OBJECTIVES :
-Know characteristics of matrices
-Apply basic operations on matrices
-Know the inverse matrices (up to 3X3)
-Solve simultaneous linear equations up to
3 variables