Types of Matrices

#Types of Matrices or Matrix:

Row Matrix

A matrix is said to be a row matrix if it has only one row, but may have any number of

columns,

e.g :    $\begin{bmatrix} 1&6&0&1&2 \end{bmatrix}$
The order of a row matrix is " 1 x n ".

Column Matrix
  A matrix is said to be a Column matrix if it has only one column, but may have any
number of rows,
e.g :

    $\begin{bmatrix} 2\\3\\0\\7 \end{bmatrix}$
The order of a column matrix is " m x 1 ".

Square Martrix
A matrix is said to be a Square matrix if number of rows is equal to the number of
column,

e.g : the matrix    $\begin{bmatrix} 1&2&3\\ 0&6&1\\ 3&4&2 \end{bmatrix}$ having 3 rows and 3 columns is a square matrix.
The order of a square matrix is " n x n " .

Note : In any given matrix   $A = \begin{bmatrix} a_{ij} \end{bmatrix}$ of order " m x n ", the elements of the principal
diagonal are   $a_{11},a_{22},a_{33},....,a_{nn}$ .

Rectangular Matrix
A matrix is said to be a rectangular matrix if the number of rows is not equal to the
number of column,

e.g : the matrix   $\begin{bmatrix} 2&3&4\\ 5&1&2\\ 3&0&-1\\ 2&1&3 \end{bmatrix}$ having 3 rows and 4 columns is a rectangular matrix.
It may be noted that a row matrix of order " 1 x n " (n ≠ 1 ) and a column matrix
of order " m x 1 " ( m ≠ 1 ) are rectangular matrix.

Zero or Null Matrix
A matrix each of whose element is zero is called a zero or null matrix.
e.g : each of the matrix

   $\begin{bmatrix} 0&0 \end{bmatrix} , \begin{bmatrix} 0&0\\ 0&0 \end{bmatrix} , \begin{bmatrix} 0&0&0\\ 0&0&0 \end{bmatrix} , \begin{bmatrix} 0&0\\ 0&0\\ 0&0 \end{bmatrix} , \begin{bmatrix} 0&0&0\\ 0&0&0\\ 0&0&0 \end{bmatrix}$
is a zero matrix. Zero matrix is denoted by " O ".

Diagonal Matrix
A square matrix is said to be a diagonal matrix, if all elements other than those occuring
in the principal diagonal are zero, i.e., if   $A = \begin{bmatrix} a_{ij} \end{bmatrix}$ is a square matrix of order " m x n ",
then it is said to be a diagonal matrix if   $a_{ij}\neq\0$ for all   $i\neq\j$ .

e.g :    $\begin{bmatrix} 3 & 0 & 0\\ 0 & 4 & 0\\ 0 & 0 & 5 \end{bmatrix} , \begin{bmatrix} 7& 0 & 0 & 0\\ 0 &3 & 0& 0\\ 0& 0 & 1 & 0\\ 0& 0 & 0 & 8 \end{bmatrix}$ are diagonal matrix.
Note : A diagonal matrix $A = [a_{ij}]_{n\times n}$ is also written as $A = diag[a_{11},a_{12},a_{13},...,a_{nn}]$ .

Scalar Matrix
A diagonal matrix is said to be a Scalar matrix if all the elements in its principal diagonal
are equal to some non-zero constant, say k
e.g : the matrix $\begin{bmatrix} -3 & 0 &0 \\ 0& -3 & 0\\ 0& 0 & -3 \end{bmatrix}$ is a scalar matrix .
Note : A Square zero matrix is not a scalar matrix.

Unit or Identity Matrix
A scalar matrix is said to be a unit or identity matrix if all of its elements in the principal
diagonal are unity. It is denoted by $I_{n}$ , if it is of order "n" .
e.g : the matrix $\begin{bmatrix} 1 & 0&0 \\ 0& 1 & 0\\ 0& 0 & 1 \end{bmatrix}$ is a Unit matrix of order 3.
Note : A square matrix $A = \begin{bmatrix} a_{ij} \end{bmatrix}$ is a unit matrix if $a_{ij} = \left\{\begin{matrix} 0, &when&i \neq j \\ 1, & when& i = j \end{matrix}\right.$

Equal Matrix
Two matrix are said to be equal if they are of the same order and if their corresponding
elements are equal.

If A is a matrix of order " m x n " and B is a matrix of order " p x r ", then A = B if
(1) m = p ; n = r ; and
(2) $a_{ij} = b_{ij}$ for all 3 x 2 and j = 1 , 2 , 3 ,........, n

Two matrix X and Y given below are not equal, since they are of different orders, namely
2 x 3 and 3 x 2 respectively.

$X = \begin{bmatrix} 7 & 1 & 3\\ 2 & 1 & 5 \end{bmatrix} , Y = \begin{bmatrix} 7 & 2\\ 1& 1\\ 3 & 5 \end{bmatrix}$
Also, the two matrix P and Q are not equal, since some elements of P are not equal to the
corresponding elements of Q.

$P = \begin{bmatrix} -1 & 3 & 7\\ 0 & 1 & 2 \end{bmatrix} , Q = \begin{bmatrix} -1 & 3 & 6\\ 0 & 2 & 1 \end{bmatrix}$

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Types of Matrices

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