UNIT – 3
MATRICES AND LINEAR SYSTEM OF EQUATIONS
INTRODUCTION:
The concept of matrix developed as result of an attempt to find concise &
simple methods for solving a system of linear equations. Matrix is not only used
to express the co-efficients of a system of linear equations. Matrix is not only
used to express the co-efficients of a system of linear equations but the ability of
matrices is much more than this experiment.
Matrix notation & operations are used in electronics spreadsheet
programs for personal computers, which are used in various fields of science &
commerce respectively, such as budgeting, sales out, cost estimation, analysis of
the results of an experimental etc. Nowadays matrices are used in mathematics,
statistics, physics, chemistry, sociology, linear programming, probability theory,
etc.
BASIC DEFINITION:
A matrix is rectangular array of elements arranged in horizontal rows &
vertical columns.
[ ]
1 −1 2
Ex: A = 3 1 0
7 9 4
NOTATION OF MATRIX:
A matrix is an ordered system of numbers arranged in formation of rows
& columns, thus a system of mn numbers, real or complex, arranged in a
rectangular arrays of m rows and n columns, written as:
[ ] ( )
a 11 a21 … .. a 1 n a 11 a 12 … .. a 1 n
Ex: a 21 a22 … .. a 2 n or a21 a 22 … .. a 2 n [aij] mxn
a 31 a32 … .. a 3 n a31 a 32 … .. a 3 n
Where I = 1,2,3,…m & j = 1,2,3,….n is caleed mxn matrix.
ORDER OF MATRIX:
A matrix having m rows & n columns called a matrix of order mxn.
3 1
If A = 2 3 then the order of matrix mxn = 3x2.
2 0
EQUALITY OF MATRIX:
[ a 11
A = a 21 a 22
a12
] [ b 11
B = b 21 b 22
b 12
]
a11 = b11, a12 = b12, a21 = b21, a22 = b22.
ROW MATRIX:
A matrix having only one row is called row matrix or row vector.
Ex: [7 3 ] , ¿
COLUMN MATRIX:
A matrix which has only one column is called column matrix.
1
Ex: 2
3
SINGLETON MATRIX:
A matrix which has only one element is called singleton matrix.
Ex: [ 4 ]
HORIZONTAL MATRIX:
When a number of rows is less than the number of column, the matrix is
called a horizontal matrix.
1 2 3
Ex: 7 1 9
VERTICAL MATRIX:
When a number of rows is more than the number of column, the matrix is
called a vertical matrix.
1 2
Ex: 8 3
6 1
SQUARE MATRIX:
A matrix which has equal number of rows & columns is called square
matrix. It is denoted by nxn order.
[ ]
1 4 5
Ex: 0 2 8
0 0 3
NULL MATRIX OR ZERO MATRIX:
If each element of a matrix is zero, the matrix is called null or zero
matrix.
[ 0 0]
Ex: 0 0
DIAGONAL MATRIX:
A square matrix is called a diagonal matrix of each of its non-diagonal
elements are zero.
[ ]
1 0 0
Ex: 0 4 0
0 0 2
TRIANGULAR MATRIX:
A square matrix in which either all the entries above the principal
diagonal or all the entries below the principal diagonal or zero.
Upper triangular matrix: A square matrix is said to be upper triangular matrix
if all the elements below the diagonal are zero.
[ ]
1 4 5
Ex: 0 2 8
0 0 3
Lower triangular matrix: A square matrix is said to be lower triangular matrix
if all the elements above the diagonal are zero.
[ ]
1 0 0
Ex: 4 2 0
5 7 3
SINGULAR MATRIX:
A square matrix A is said to be singular if |A| = 0.
[ ]
1 2 4
Ex: 3 2 3 Here |A| = 0
2 4 8
TRANSPOSE OF MATRIX:
If the rows & columns of a matrix are interchanged the new matrix
formed is called the transpose of matrix. It is written as A = AT .
[ ] [ ]
1 2 4 1 3 2
T
Ex: 3 2 3 A = 2 2 4
2 4 8 4 3 8
