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