MATRICES

Matrix: A system of mn numbers real (or) complex arranged in the form of an ordered set

of 'm' rows, each row consisting of an ordered set of 'n' numbers between [] (or) () (or) || |||

is called a matrix of order m xn.

Eg:

Column Matrix: A Matrix having only one column is called a "Column Matrix".

= [aij]mxn

mxn

Order of the Matrix: The number of rows and columns represents the order of the matrix. It is denoted by mxn, where m is number of rows and n is number of columns.

Types of Matrices:

Row Matrix: A Matrix having only one row is called a "Row Matrix".

Eg: [1 2 3

Eg: 1

Null Matrix: A=

denoted by Omxn.

To 0 0

Eg: 02x3= 0 0

where I<i<m, 1<j<n.

[a] such that a0i and j. Then A is called a "Zero Matrix". It is

Rectangular Matrix: If A-[a, and mn then the matrix A is called a "Rectangular

Matrix"

Eg: [1 2 -1 2 3 4 is a 2x3 matrix

Square Matrix: If A-[a] and m n then A is called

Eg:|2 1 1

2 2 is a 2x2 matrix

a "Square Matrix".

Lower Triangular Matrix: A square Matrix Axn [a] is said to be lower Triangular

of 0 ifiji.e. if all the elements above the principle diagonal are zeros. [4 0 0

Eg:

2

7 3

0

6

is a Lower triangular matrix

Upper Triangular Matrix: A square Matrix [] is said to be upper triangular of

"=0 ifij. i.e. all the elements below the principle diagonal are zeros.

[1 3 8

Eg: 0 4 -5

0 0 2

is an Upper triangular matrix

Triangle Matrix: A square matrix which is either lower triangular or upper triangular is

called a triangle matrix.

Principal Diagonal of a Matrix: In a square matrix, the set of all aij, for which i j are

called principal diagonal elements. The line joining the principal diagonal elements is called

principal diagonal.

Note: Principal diagonal exists only in a square matrix.

Diagonal elements in a matrix: A= [a], the elements a; of A for which i=j.

i.e. ai, az2....ann are called the diagonal elements of A

2 4 5 Eg: A=

diagonal elements are 1, 5, 9.

Diagonal Matrix: A Square Matrix is said to be diagonal matrix, if a,,=0 for ij i.e. all

the elements except the principal diagonal elements are zeros. Note: 1. Diagonal matrix is both lower and upper triangular.

2. If di, d2...d are the diagonal elements in a diagonal matrix it can be

represented as diag [d, d d ]

300

Eg: A diag (3,1,-2) 01 0

00-2

Scalar Matrix: A diagonal matrix whose leading diagonal elements are equal is called a

"Scalar Matrix".

Eg: A= 2 0 07 0 2 0

0 0 2

Unit/Identity Matrix: If A = [a, such that a-1 for i= j, and a-0 for i then A is

called a "Identity Matrix" or Unit matrix. It is denoted by I

Eg: b₂ = =

DEPARTMENT OF HUMANITIES & SCIENCES CMRCET (EAMCET CODE: MLRD)

1 0 0

130 1 0

0 0

Trace of Matrix: The sum of all the diagonal elements of a square matrix A is called Trace

of a matrix A, and is denoted by Trace A or tr A.