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
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
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
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
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
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
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
