introduction of matrices

MATRICES

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𝑎11 𝑎12 𝑎13

𝑎21 𝑎22 𝑎23

𝑎31 𝑎32 𝑎33

Introduction of Matrices

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Matrices 2


S. No. Topic Page No.

1 Definition of Matrices 3-4

2 Notation of Matrices 5

3 Order of Matrices 6

4 Types of Matrices 7-11

5 Trace of Matrices 12

6 Equality of Two Matrices 13

7 MCQ‟s for Exercise 14-15

8 Answer Key 16

Index



A matrix is a rectangular array of numbers. In other words, a set of „m‟, „n‟ numbers arranged in the form of rectangular array of m rows and n column is called m×n matrix read as „m‟ by „n‟ matrix.

A =

a11 a12 a13 … … … . .a21 a22 a23 … … … . .a31 a32 a33 … … … . .

m×n

We can understand matrix by the fig. shown above, which shows a mess of vertical and horizontal lines. The crossing points of vertical and horizontal lines are the position of elements of matrix. In above fig there are 9 crossing points which can be find out by multiplying no. of horizontal and vertical lines i.e. 3 × 3 = 9.

Matrices 3


Horizontal lines

Vertical lines

Definition of Matrix



The numbers a11, a12 ……..etc are called elements of the matrix A. „m‟ is the numbers of rows and „n‟ is number of column.

e.g. A= 2 3 45 4 62 4 3

3×3

Matrices 4




• Matrices are denoted by capital letters A, B, C or X , Y , Z etc.

• Its elements are denoted by small letters a, b, c ….. etc.

• The elements of the matrix are enclosed by any of the brackets i.e. , , .

• The position of the elements of a Matrix is indicated by the subscripts attached to the element. e.g. 𝑎13 indicates that element „a‟ lies in first row and third column i.e. first subscript denotes row and second subscript denote column.

Matrices 5


𝑎1 3 Element Indicate Row

Indicate Column

Notation of Matrices



• The number of rows and columns of a matrix determines the order of the matrix.

• Hence , a matrix , having m rows and n columns is said to be of the order m × n ( read as „m‟ by „n‟).

• In particular, a matrix having 3 rows and 4 columns is of the order 3 × 4 and it is called a 3 × 4 matrix e.g.

A= 2 3 54 6 7

is a matrix of order 2 × 3 since there are two rows and three

columns.

A = 3 5 7 is a matrix of order 1 × 3 since there are one row and three columns.

A =568

is a matrix of order 3 × 1 since there are three rows and one

column.

Matrices 6


Order of Matrices



Matrices 7

RectangularMatrix

Square Matrix

Diagonal

Matrix

Scalar Matrix

Unit Matrix

Null

Matrix

Row Matrix

Column Matrix

Upper Triangular

Matrix

Lower

Triangular Matrix

Sub Matrix

Types of Matrices




Rectangular Matrix : A matrix in which the number of rows and columns are not equal is called a rectangular matrix e.g. ,

A= 2 4 54 6 7

2 × 3 is rectangular matrix of order 2×3

Square Matrix : A matrix in which the number of rows is equal to the number of columns is called a square matrix e.g.

𝐴 =2 3 43 2 57 8 5

Note : The elements 2, 2 , 5 in the above matrix are called diagonal elements and the line along which they lie is called the principal diagonal

Matrices 8


Principal Diagonal



Diagonal Matrix : A square matrix in which all diagonal elements are non- zero and all non-diagonal elements are zeros is called a diagonal matrix e.g.

𝐴 =2 0 00 2 00 0 5

is a diagonal matrix of 3 × 3

Scalar Matrix : A diagonal matrix in which diagonal elements are equal (but not equal to 1), is called a scalar matrix e.g.

𝐴 =2 0 00 2 00 0 2

is a scalar matrix of 3 × 3

Identity (or Unit) Matrix : A square matrix whose each diagonal element is unity and all other elements are zero is called and Identity (or Unit) Matrix. An Identity matrix of order 3 is denoted by I3 or simply by I.

Matrices 9




e.g.

I3 =1 0 00 1 00 0 1

is a unit matrix of order 3

Null (Zero) Matrix : A matrix of any order (rectangular or square) whose each of its element is zero is called a null matrix (or a Zero matrix) and is denoted by O. e.g.

O =0 00 0

and O =0 0 00 0 0

are null matrices of order 2 × 2 and 2 × 3

respectively.

Row Matrix : A matrix having only one row and any number of columns is called a row matrix (or a row vector) e.g.

𝐴 = 1 2 3 is a row matrix of order 1 × 3

Matrices 10




Column Matrix : A matrix having only one column and any number of rows is called a column matrix (or a column vector) e.g.

A =136

is a column matrix or order 3 × 1

Upper Triangular and Lower Triangular Matrix: A square matrix is called an upper triangular matrix if all the elements below the principal diagonal are zero and it is said to be lower triangular matrix if all the elements above the principal diagonal are zero e.g.

𝐴 =2 3 40 1 50 0 7

, 𝐵 =5 0 08 7 04 3 1

Sub Matrix : A matrix obtained by deleting some rows or column or both of a given matrix is called its sub matrix. e.g.

𝐴 =2 3 45 1 53 9 7

, Now 2 33 9

is a sub matrix of given matrix A. The sub

matrix obtained by deleting 2nd row and 3rd column of matrix A.

Matrices 11


LTM UTM



The sum of all the elements on the principal diagonal of a square matrix is called the trace of the matrix. It is denoted by tr. A .

If A=

a11 a12 a13

a21 a22 a23

a31 a32 a33

then

trace of A = tr.A = 𝑎11 + 𝑎22 + 𝑎33

e.g. A =2 3 45 1 53 9 7

, then

trace of A = tr. A =2+1+7=10

Matrices 12


Trace of Matrix



Two matrices „A‟ and „B‟ are said to be equal if only if they are of the same order and each element of „A‟ is equal to the corresponding element of „B‟. Given

A =

a11 a12 a13……..

a21 a22 a23……..

a31 a32 a33……..

m×n and B =

b11 b12 b13………

b21 b22 b23………

b31 b32 b33………

x×y

The above two matrices will be equal if and only if

No. of rows and column of A and B are equal i.e. m=x and n=y.

Each element of A is equal to the corresponding element of B i.e. a11=b11, a12 = b12, a13 = b13 ……… and so on

Let 𝐴 =𝑎 𝑏𝑐 𝑑

and 𝐵 =3 45 6

, according to the condition of equality of

matrices both matrices have same order i.e. 2 × 2 but the 2nd necessary condition is a=3, b=4, c=5 , d= 6

Matrices 13


Equality of Two Matrices



1. If in a given matrix the rows are 2 and the column are 3 then maximum no. of elements in this matrix are

A) 4 B) 6 C) 8 D) 10

2. Element a32 is belongs to which row and column in a given matrix A

A) 2nd row, 3rd column

B) 3rd row , 2nd column

C) can’t decide

D) None of the above

3. What is the order of given matrix 2 4 57 5 79 0 3

5 78 12 4

A) 3 × 5 B) 5 × 3 C) 3 × 3 D) 5 × 5

4. A matrix which do not have equal no. of rows and columns called

A) Square Matrix B) Triangular Matrix C) Scalar Matrix D) Rectangular Matrix

5. A matrix in which the rows and columns are equal in no. is called

A) Square Matrix B) Triangular Matrix C) Scalar Matrix D) Rectangular Matrix

6. In a Diagonal matrix all the elements on the principal diagonal can’t be equal to

A) 0 B) 1 C) 2 D) 3

Matrices 14


MCQ’s for Excercise



7. In a scalar matrix all the diagonal elements are

A) Equal to one B) Equal but greater than one C) Not equal D) None of the above

8. In a Unit matrix

A) All element are zero B) All elements are one C) Diagonal elements zero rest are one D) Diagonal elements are one rest are zero

9. In a given matrix all elements below principal diagonal are zero , the matrix is

A) Upper triangular matrix B) Lower triangular matrix C) Diagonal Matrix D) Scalar Matrix

10. In a given matrix all elements above principal diagonal are zero , the matrix is

A) Upper triangular matrix B) Lower triangular matrix C) Diagonal Matrix D) Scalar Matrix

11. Given A =

a11 a12 a13

a21 a22 a23

a31 a32 a33

then tr. A will be

A) 𝑎11 + 𝑎12 + 𝑎13 B) 𝑎11 + 𝑎22 + 𝑎33 C) 𝑎31 + 𝑎22 + 𝑎13 D) 𝑎21 + 𝑎22 + 𝑎23

12. Which one is true about the necessary condition for equality of two matrices

A) Order of two matrices are same

B) Corresponding elements of the two matrices are same

C) Both A and B

D) None of the above

Matrices 15




Matrices 16


S.No. 1 2 3 4 5 6

Ans. B B A D A A

S.No. 7 8 9 10 11 12

Ans. B D A B B C

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introduction of matrices

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