Let us consider the following system of equations in three unknowns (variables) x, y and z.
x + y + z = 10
2x –z = – 3
3x + y + 2z = 17
If we arrange the coefficients of x, y and z in the order in which they occur in the given equations
and enclose them in brackets, we get the following arrangement:
Each arrangement of the kind given by the above representation of the system of Linear
equations is called a MATRIX. The numbers which constitute the matrix are called its entries or
A matrix is not just a collection of elements, but it is a collection in which each array has a very
definite position i.e. it is an ordered array of elements.
A set of mnelements arranged in the form of an ordered set of mrows, each row consisting of an
ordered set of n elements is called an m × n matrix. It is read as ‘m by n matrix”.
ELEMENTS OF A MATRIX
The numbers occurring in the rectangular array (matrix) are called the elements of a matrix.
Each of the mn numbers of an m × n matrix is called an element of the matrix. The element
lying in the first row and first column is known as leading element.
ORDER OF A MATRIX
A matrix mby nis said to be of order m × n, if it has m rows and n columns.
REPRESENTATION OF A MATRIX
MATRIX OVER A FIELD
If all the elements of an m× nmatrix belong to a field F, then the matrix is said to be defined over
the field F. In particular, if all the elements of a matrix belong to the field of real numbers, then
the matrix is said to be real. The elements of the field F are called scalars.
DIAGONAL ELEMENTS OF A MATRIX
An element a ij of the matrix A = [aij]is said to be diagonal element if i = ji.e., the elements
a11,a22,a33 ……are the diagonal elements.
The line along which the diagonal elements lie is called the principle diagonal or simply diagonal
of a matrix.
TYPES OF MATRICES
A. ROW MATRIX/ROW VECTOR
A matrix having only one row is called a row matrix or a row matrix.
B. COLUMN MATRIX/COLUMN VECTOR
A matrix having only one column is called a column matrix or column vector.
C. RECTANGULAR MATRIX
A matrix in which number of rows is not equal to the number of columns or vice-versa is called a
D. SQUARE MATRIX
A matrix having equal number of rows and columns is called a square matrix. If the matrix A has
n rows and n columns, it is said to be a square matrix of order n.
E. ZERO MATRIX/NULL MATRIX
A matrix square or rectangular, each of whose element is zero is called a zero matrix or
null matrix. A zero matrix of order m × n is denoted by O m×n .
F. DIAGONAL MATRIX
A square matrix A = [a ij ]is said to be a diagonal matrix, if all its elements except the diagonal
elements are zero. Thus a matrix A = [a ij ]is said to be diagonal matrix if