matrix

DAV INSTITUTIONS ODISHAZONE 1
DEPT. OF MATHEMATICS

LEARNING OBJECTIVES:-
 Cost estimation, Sale projection and factory
problems can be solved by using matrix.
 Expressing in vector form
 Expressing day to day life problems in matrix form
 Matrix notation and operations are used in
electronic spreadsheet, advanced statistics.
 Expressing simultaneous linear equations in
matrix form.

Defination of matrix:-
3
● A matrix is an ordered rectangular array of numbers
that represent some data ( Plural = matrices)
● A matrix on its own has no value – it is just a
representation of data
● Could be data associated with manufactured quantity in
a factory, speed of a rocket etc
● Forms the basis of computer programming
● A matrix is used in solving equations that represent
business problems

Types of matrix :-
 Row matrix: it having only one row Ex
 Column matrix: it having only one column Ex
 Zero matrix: A matrix is called a zero matrix if all the entries are 0 Ex
 Square matrix: if number of rows is equal to number of columns
 Note: if number of rows = no of number columns =n, is called square matrix of order n or order n
 order 2 order 3

Types of matrix :-
 Diagonal matrix: A square matrix is called diagonal matrix, if all of its non-
diagonal elements are zero.
 EXAMPLE
 Scalar matrix: A square matrix is called scalar matrix if diagonal elements
are same and other are “0”
 EXAMPLE
 Identity/ unit matrix : A square matrix is identity if diagonal entries are 1
and other are 0.

REPRESENTATION OF MATRIX
aij =
6
element in row‘i’ and column‘j’,where‘a’ is
an element in the matrix
Eg: a 23 = element in 2nd row and 3rd column = 9
A = [aij]MxN

Examples of Matrices
2 4
5 7
2 3 6
72 3 9
7 9 11 5
9 0 3 6
This is an example of a 2 x 2 matrix
What is a12
What is the dimension or der
of this Matrix?
What is a12
What is the dimension or order
of this Matrix?
What is a12 ?
7

Addition operation on Matrices
2
6
7
9
2
8
45 72
3 0
9 10
A
40 7
6 1
7 2
B
(2+40) (45+7) (72+9) 47 52 81
(6+6) (3+1) (0+2) 12 4 2
(7+7) (9+2) (10+8) 14 11 18
Only Matrices of the same order(comparable) can be added!!
Rule 1: A + B =B + A
8

Question Set 1
1. Add the following matrices:
32 4 60
29 2 4
21 65 7
22 5
10 8
9 7
8
12
2
2. Subtract the following matrices:
18 26 12 7 2 15
10 11 12 13 3 5
8 10 16 5 8 9
9

Multiplication of a matrix by a scalar
10
If K is any number and A is a givenmatrix,
Then KA is the matrix obtained by
multiplying each element of A byK.
K is called‘Scalar’. Eg: if K = 2
2 4 5 4 8 10
A= 1 3 2 KA = 2 6 4
2 5 1 4 10 2

MULTIPLICATION OF MATRICES
 The product AB of two matrices A and B is defined, if the number of
columns of A is equal to the number of B.
 If AB is defined then BA need not be defined . In particular both A and B
are square matrices of same order then AB and BA are defined.
 In general AB≠BA
 Observation : Two non zero matrices multiplication is zero matrix

Multiplication of Matrices - 2
2 3 1
4 3 2
4 2
1 0
5 2
(2x4+3x1+1x5)
(4x4+3x1+2x5)
16
29
13
15
2 x 3 matrix
A B
3 x 2 matrix
(2x2+3x0+1x2)
(4x2+3x0+2x2)
A x B
2 x 2 matrix
12

Multiplication of Matrices: - 1
1 3 5 2 1 3
2 4 2 4 5 2
2 5 6 6 2 3
(1x2 + 3x4+5x6)
(2x2 + 4x4+2x6)
(1x1+ 3x5+5x2)
(2x1+ 4x5+2x2)
(1x3+3x2+5x3)
(2x3+4x2+2x3)
(2x2 + 5x4+6x6) (2x1+ 5x5+6x2) (2x3+5x2+6x3)
25
32
58
26
26
39
24
20
34
14

Multiplication of Matrices - 3
2
4
3
3
1
2
4
1
5
2
0
2
(4x2+2x4)
(1x2+0x4
(5x2+2x4)
(4x3+2x3)
(1x3+0x3)
(5x3+2x3)
(4x1+2x2)
(1x1+0x2)
(5x1+2x2)
16 18 8
2 3 1
18 21 9
3 x 2 matrix
A
2 x 3 matrix
B
B xA
3 x 3 matrix
Rule 2: Ax B B xA 15

Question Set 1
16
3. Multiply the following matrices:
2 3 4
0 10 3
1 0 1
1 0 5
5 6 9
1 2 0
4. 1 0
2 1
2 3 1
1 2 10
5. 3
2
4
1
2
0
2 3
4 2
1
2
Is it possible to compute No.5?! No!Why?

Transpose of a Matrix
NJ Jaissy 17
● Matrix formed by interchanging rows and
columns of A is called A transpose(A’)
Q.Verify (A + B)’= A’ + B’
Q. Verify (A B)’ = B’ xA’

Question Set 1
18
6. Find the transpose of the following matrices and
verify that (A+B)’ = A’ +B’
A = 1 2 9 B= 2 3 4
4 3 6 1 8 6
Hint: FindA+B, (A+B)’,A’ and B’ and verify
7. If D is a matrix where first row = number of table fans
and second row = number of ceiling fans factories A and
B make in one day.If a week has 5 working days compute
5A. What does 5A represent?
D = 10
30
20
40

Question Set 1
9. T
woshops have the stock of large, medium and small sizes of a
toothpaste.The number of each size stocked is given by the matrixA where
Large
A = 150
90
Medium Small
240 120
300 210
shop no.1
shop no 2
The cost matrix B of the different size of the toothpaste is given by
B = Find the investment in toothpaste by each shop
Cost
14
10
6
Answer:
17
3820 -- Investment by shop no 1 5520
-- InvestmNJeJanissytby shop no 2

Question Set 1
20
8.
A =
For the matrix
4 5 6
and B =
7 9
2 1 3 10 2
-5 2 2
Multiply by the Matrix I =
1 0 0 1 0
0 1 0 0 1
0 0 1
What is A.I and I.B ?

Identity Matrix
21
● If you were to multiply ‘a’ by ‘1’, you would get ‘a’ .
Eg: 2 x 1 =2x1 =2
● The ‘identity’ matrix (i) is the equivalent of ‘1’ in basic math
If A is a matrix and I is an identity Matrix,
● Then A x I =A and I xA =A. Identity Matrices
1 0 1 0 0
0 1 0 1 0
0 0 1
ThenA x I =A and I xA =A
a b 1 0 a+0 0+b a b
c d 0 1 c+0 0+d c d

To find inverse by using elementary Row
transformations.
Step 1: Write A = IA
Step 2: Apply various row operations
on left hand side and apply same
operations to I on right side but not to
A on right side.
Step3: From step 2 we get a new
matrix equation I = BA . Hence B = A-1
.

To find inverse by using elementary Column
transformations.
Step 1: Write A = AI
Step 2: Apply various Column
operations on left hand side and apply
same operations to I on right side but
not to A on right side.
Step3: From step 2 we get a new
matrix equation I = AB . Hence B = A-1
.

Inverse of a Matrix
NJ Jaissy 27
In basic math: 2 2 = 1 and 1/2 x 2 = I.
Dividing 2 by two is the same as multiplying 2 by 1/2 .The
net result is 1.
A similar concept is the‘inverse’ of a matrix. If A is a
matrix,then A is the inverse such
that AxA = I (identity matrix)
If A has an inverse(A ) then A is said to be ‘invertible’
A.A=A.A = I

KEY POINTS
 A matrix is an ordered rectangular array of numbers or
functions.
 A matrix having m rows and n columns is called a matrix
of order mxn
 A is a diagonal matrix if its non diagonal elements are
zero.
 A is a identity matrix if diagonal elements are 1 and non
diagonal elements are 0
 A is zero matrix if all elements are zero.
 Matrix addition is commutative ,associative over same
order. A+B= B+A, (A+B)+C=A+(B+C)

KEY POINTS
 k (A+B)= kA+kB , k is constant A,B are of same order
 If order of first matrix A is m and that of B is n, then A.B is possible of order m
 Matrix multiplication is not commutative.
 A matrix is symmetric if
 If a square matrix is invertible if detA≠0, or singular matrix
 if detA =0,inverse of a square matrix doesn’t exist.or non singular matrix
ist

matrix

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