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.
For example,
A = (5 3 4 1) and B = (–3 0 5 )
are row matrices of orders 1 x 4 and 1 x 3 respectively.
In general,
A = (a_{ij})_{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.
For example,
are column matrices of orders 2x1 and 3x1 respectively.
In general,
A = [a_{ij}]_{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.
For example,
are square matrices of orders 2 and 3 respectively.
In general,
A = [a_{ij}]_{mxm}
is a square matrix of order m.
The elements a_{11}, a_{22}, a_{33} ....... a_{mm} 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.
For example,
are diagonal matrices of orders 2 and 3 respectively.
In general,
A = [a_{ij}] _{mxm }
said to be a diagonal matrix if a_{ij} = 0 for all i ≠ j.
Note :
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.
For example,
are scalar matrices of orders 2 and 3 respectively.
In general,
A = [a_{ij}]_{ mxm}
is said to be a scalar matrix if
a_{ij} = 0, when i ≠ j
a_{ij }= k. when i = j
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 I_{n}. For example,
are unit matrices of orders 2 and 3 respectively.
In general, a square matrix A = [a_{ij}] _{nxn} is a unit matrix if
a_{ij} = 1 if i = j
a_{ij }= 0 if i ≠ j
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.
For example,
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 A^{T} (read as A transpose).
For example,
In general, if A = [a_{ij}]_{ mxn} then A^{T} = [b_{ij}]_{nxm }where
b_{ij} = a_{ij}
for i = 1, 2,......n and j = 1, 2,...... m.
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