# Types Of Matrices

A matrix is nothing but a rectangular array of numbers or functions. It has two dimensions. These two dimensions are rows and the columns. The number of the rows are represented by m and the number of columns are represented by n. The matrices are basically categorized on the basis of the value of their elements, their order, number of rows and columns etc. Now, using different conditions, we shall look at the types of matrices with definition and examples. These are the different types of matrices:

• Square Matrix: When m = n for any matrix, it is known as a Square matrix. It is easy to understand this by a little hint of geometry. What happens to a rectangle when its two dimensions become equal to each other? It becomes a square.

The order of the matrix hence becomes m × m or n × n. Instead of calling it as an m by m matrix, it is known as a matrix of order m.

In general, a square matrix A , of order m , is represented as:

A = $[a_{ij}]_{m×m}$

e.g $P = \begin{bmatrix} 4 & 7\\ 9 & 13 \end{bmatrix}$

Q =$\begin{bmatrix} 2 & 1 & 13\cr -5 & -8 & 0\cr 14 & -7 &9 \end{bmatrix}$

The order of P and Q is 2 ×2 and 3 × 3 respectively.

What about the matrix which has a single element? Can we call it a square matrix? The answer is yes. We can call it as a square matrix because the order is 1 × 1 i.e. single row and column.

[5],[9],[-2015] and [9$\sqrt{13}$] are all square matrices.

• Row Matrix: If for any matrix,m = 1 , then it is known as a row matrix. It is called so because it has only one row and The order of a row matrix will hence be 1 × n . A row matrix is generally represented as:

B = [ $b_{ij} ]_{1 × n}$

For e.g.

P = [ -4 -21 -17 ]

Q = [ 5 13 9 -2$2\sqrt{5}$ -49 ]

The order of P and Q is 1 × 3 and 1 × 5 respectively.

• Column Matrix: Just like the row matrices had only one row, column matrices have only one column. Thus, the value of for a column matrix will be 1. Hence, the order is m × . The general form of a column matrix is given by:

$~~~~~~~~~~$ B = [$b_{ij} ]_{m × 1}$

For e.g.

$~~~~~~~~~~$P =$\begin{bmatrix} 2 \cr 7\cr -17 \end{bmatrix}$

$~~~~~~~~~~$Q =$\begin{bmatrix} -1 \cr -18\cr -19\cr 9\cr 13 \end{bmatrix}$

In the above example , P and Q are 3 ×1 and 5 × 1 order matrices respectively.

• Diagonal Matrix: Going by the name, something has got to be special about diagonal of this matrix. And that special thing is, all the non-diagonal elements of this matrix are zero. That means only the diagonal has non-zero elements. There are two important things to note here. A diagonal matrix is always a square matrix. And the diagonal elements are characterized by this general form:$b_{ij}$ where i = j. This means that a matrix can have only one diagonal. The general form of a diagonal matrix is:

A = [ $a_{ij} ]_{m× m}$ where $a_{ij}$ = 0 when i ≠ j.

Following are some examples that depict diagonal matrices:

P = [9]

Q =$\begin{bmatrix} 9 & 0 \cr 0 & 13 \end{bmatrix}$

$R = \begin{bmatrix} 4 & 0 & 0\\ 0 & 13 & 0 \\ 0 & 0 & -2 \end{bmatrix}$

In the above examples, P, Q, and R are diagonal matrices with order 1 × 1, 2 × 2 and 3 × 3 respectively. When all the diagonal elements of a diagonal matrix are the same, it goes by a different name, scalar matrix. A scalar matrix is represented as:

B = $b_{ij}]_{m × m}$

where $b_{ij}$ = 0 when i ≠ j

and $b_{ij}$ = k (constant) , when i = j

Some examples of scalar matrices are:

P =$\begin{bmatrix} 3 & 0 \cr 0 & 3 \end{bmatrix}$

$Q = \begin{bmatrix} \sqrt{5} & 0 & 0\\ 0 & \sqrt{5} & 0 \\ 0 & 0 & \sqrt{5} \end{bmatrix}$

What if all the diagonal elements are equal to 1? That will still be a scalar matrix and obviously a diagonal matrix. It has got a special name. It is known as the identity matrix. It is represented by:

C = $[c_{ij}]_{m × m}$

where $c_{ij}$ = 0 when i ≠ j

and $c_{ij}$ = k (constant) , when i = j

An identity matrix is commonly written as where is the order of the matrix. Some examples of identity matrix are:

I1 = [1]

$I_3$ =$\begin{bmatrix} 1 & 0 & 0 \cr 0 & 1 & 0 \cr 0 & 0 & 1 \end{bmatrix}$

So, we have three conclusions:

• All identity matrices are scalar matrices
• All scalar matrices are diagonal matrices
• All diagonal matrices are square matrices

But the converse is not true in any of the above cases.

Zero Matrix: This is simplest of all. If all the elements of a matrix are equal to zero, the matrix is known as a zero matrix or null matrix. It is denoted by O (Big O). Some examples are:

P = [0]

Q =$\begin{bmatrix} 0 \cr 0 \end{bmatrix}$

$R = \begin{bmatrix} 0 & 0 & 0\\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{bmatrix}$

Those were few different types of matrices with definitions and examples. We also saw some special types of matrices. The most useful of all these matrices are Identity matrices. To learn more along with algebra of matrix, visit www.byjus.com and explore more.

#### Practise This Question

Sum of an upper triangular matrix and lower triangular matrix can result in a triangular matrix.