Row matrix :
A matrix is said to be a row matrix, if it has only one row. A row matrix is also called as a row vector.
A = (5 3 4 1) and B = (–3 0 5 )
are row matrices of orders 1 x 4 and 1 x 3 respectively.
A = (aij)1xn
is a row matrix of order 1xn.
Column matrix :
A matrix is said to be a column matrix, if it has only one column. It is also called as a column vector.
are column matrices of orders 2x1 and 3x1 respectively.
A = [aij]mx1
is a row matrix of order mx1.
Square matrix :
A matrix in which the number of rows and the number of columns are equal is said to be a square matrix.
are square matrices of orders 2 and 3 respectively.
A = [aij]mxm
is a square matrix of order m.
The elements a11, a22, a33 ....... amm are called principal or leading diagonal elements of the square matrix A.
Diagonal matrix :
A square matrix in which all the elements above and below the leading diagonal are equal to zero, is called a diagonal matrix.
are diagonal matrices of orders 2 and 3 respectively.
A = [aij] mxm
said to be a diagonal matrix if aij = 0 for all i ≠ j.
Some of the leading diagonal elements of a diagonal matrix may be zero.
Scalar matrix :
A diagonal matrix in which all the elements along the leading diagonal are equal to a non-zero constant is called a scalar matrix.
are scalar matrices of orders 2 and 3 respectively.
A = [aij] mxm
is said to be a scalar matrix if
where k is constant.
Unit matrix :
A diagonal matrix in which all the leading diagonal entries are 1 is called a unit matrix. A unit matrix of order n is denoted by In. For example,
are unit matrices of orders 2 and 3 respectively.
In general, a square matrix A = [aij] nxn is a unit matrix if
A unit matrix is also called an identity matrix with respect to multiplication. Every unit matrix is clearly a scalar matrix. However a scalar matrix need not be a unit matrix. A unit matrix plays the role of the number 1 in numbers.
Null matrix or Zero-matrix :
A matrix is said to be a null matrix or zero-matrix if each of its elements is zero. It is denoted by O.
are null matrices of order 2x3 and 2x2.
(i) A zero-matrix need not be a square matrix.
(ii) Zero-matrix plays the role of the number zero in numbers.
(iii) A matrix does not change if the zero-matrix of same order is added to it or subtracted from it.
Transpose of a matrix :
Definition The transpose of a matrix A is obtained by interchanging rows and columns of the matrix A and it is denoted by AT (read as A transpose).
In general, if A = [aij] mxn then AT = [bij]nxm where
bij = aij
for i = 1, 2,......n and j = 1, 2,...... m.
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