1. Definition : Rectangular array of m n numbers. Unlike determinants it has no value.
Abbreviated as : A = [aij] 1 £ i £ m ; 1 £ j £ n, i denotes the row and j denotes the column is called a matrix of order m x n.
2. Special Type Of Matrices :
(a) Row matrix : A = [a11, a12, ..........a1n] having one row. (1 x n ) matrix. (or row vectors)
(b) Column Matrix (or column vectors ) A =
having one column. (m x 1 ) matrix
(c) Zero or Null Matrix : (A = Omxn)
An m x n matrix all whose entries are zero. A=
is a 3 x 2 null matrix & B =
is 3 x 3 null matrix
(d) Horizontal Matrix : A matrix of order m x n is a horizontal matrix if n > m .
(e) Vertical Matrix : A matix of order m x n is a vertical matrix if m>n.
(f) Square Matrix : (Order n)
If number of rows = number of columns Þ a square matrix.
Note (i) In a square matrix the pair of elements aij & aji are called Conjugate Elements.
e.g. in the matrix
a21 and a12 are conjugate elements.
(ii) The eleents a11, a22, a33, ................... ann are called Diagonal Elements. The line along which the diagonal elements lie is called ‘’Principal or Leading” diagonal. The qty. åaII = trace of the matrix written as , (tr A)
Note :
(i) Minimum number of zeros in an upper triangular an upper triangular matrix of order
n = n(n–1)/2
(ii) Number of zeros in a diagonal matrix unit matrix of order n = n (n–1) “It is to be noted that with every square matrix there is a correspoinding determinat formed by the elements of A in the same order. “ If |A |¹ 0 then A is called a singular matrix and if |A| then A is called a non-singular matrix.
By interchanging rows/columns of an identity matrix the matrix is transformed to an elementary matrix. The operation E13 for an identity matrix (I3) Þ interchanging first & third row the resulting matrix is :
3. Equality Of Matrices :
The matrices A = [aij] & B= [bij] are equal if ,
(i) both have the same order.
(ii) aij = bij for each pair of i & j .
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