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.
