 # Determine The Order Of Matrix

Before we determine the order of matrix, we should first understand what is a matrix. Matrices are defined as a rectangular array of numbers or functions. Since it is a rectangular array, it is 2-dimensional. Basically, a two-dimensional matrix consists of the number of rows (m) and a number of columns (n). The order of matrix is equal to m x n (also pronounced as ‘m by n’).

Order of Matrix = Number of Rows x Number of Columns

See the below example to understand how to evaluate the order of the matrix. Also, check Determinant of a Matrix. In the above picture, you can see, the matrix has 2 rows and 4 columns. Therefore, the order of the above matrix is 2 x 4. Now let us learn how to determine the order for any given matrix.

## How to determine the order of matrix?

Let us take an example to understand the concept here.

$$\begin{array}{l} A =\left[ \begin{matrix} 3 & 4 & 9\cr 12 & 11 & 35 \cr \end{matrix} \right] \end{array}$$

$$\begin{array}{l} B =\left[ \begin{matrix} 2 & -6 & 13\cr 32 & -7 & -23 \cr -9 & 9 & 15\cr 8 & 25 & 7\cr \end{matrix} \right] \end{array}$$

The two matrices shown above A and B. The general notation of a matrix is given as:

$$\begin{array}{l} A = [a_{ij}]_{m × n} \end{array}$$
, where
$$\begin{array}{l} 1 ≤ i ≤ m , 1 ≤ j ≤ n \end{array}$$
and
$$\begin{array}{l}i , j \in N \end{array}$$

You can see that the matrix is denoted by an upper case letter and its elements are denoted by the same letter in the lower case.

$$\begin{array}{l} a_{ij} \end{array}$$
represents any element of matrix  which is in
$$\begin{array}{l} i^{th}\end{array}$$
row and
$$\begin{array}{l} j^{th} \end{array}$$
column. Similarly,
$$\begin{array}{l} b_{ij} \end{array}$$
represents any element of matrix B.

So, in the matrices given above, the element

$$\begin{array}{l} a_{21} \end{array}$$
represents the element which is in the
$$\begin{array}{l} 2^{nd} \end{array}$$
row and the
$$\begin{array}{l} 1^{st} \end{array}$$
column of matrix A.

∴a21 = 12

Similarly,

$$\begin{array}{l} b_{32} = 9 , b_{13} = 13 \end{array}$$
and so on.

Can you write the notation of 15 for matrix B ?

Since it is in

$$\begin{array}{l} 3^{rd} \end{array}$$
row and 3rd column, it will be denoted by
$$\begin{array}{l} b_{33} \end{array}$$
.

If the matrix has

$$\begin{array}{l} m \end{array}$$
rows and
$$\begin{array}{l} n \end{array}$$
columns, it is said to be a matrix of the order
$$\begin{array}{l}m × n\end{array}$$
. We call this an m by n matrix. So,  A is a 2 × 3  matrix and B is a 4 × 3  matrix. The more appropriate notation for A and B respectively will be:

$$\begin{array}{l} A =\left[ \begin{matrix} 3 & 4 & 9\cr 12 & 11 & 35 \cr \end{matrix} \right]_{2 × 3} \end{array}$$

$$\begin{array}{l} B =\left[ \begin{matrix} 2 & -6 & 13\cr 32 & -7 & -23 \cr -9 & 9 & 15\cr 8 & 25 & 7\cr \end{matrix} \right]_{4 × 3} \end{array}$$

So, if you have to find the order of the matrix, count the number or its rows and columns and there you have it.

 Note: It is quite fascinating that the order of matrix shares a relationship with the number of elements present in a matrix. The order of a matrix is denoted by a × b, and the number of elements in a matrix will be equal to the product of a and b.

### Number of Elements in Matrix

In the above examples, A is of the order 2 × 3. Therefore, the number of elements present in a matrix will also be 2 times 3, i.e. 6.

Similarly, the other matrix is of the order 4 × 3, thus the number of elements present will be 12 i.e. 4 times 3.

This gives us an important insight that if we know the order of a matrix, we can easily determine the total number of elements, that the matrix has. The conclusion hence is:

If a matrix is of  m × n  order, it will have mn elements.

### But is the converse of the previous statement true?

The converse says that: If the number of element is mn, so the order would be m × n. This is definitely not true. It is because the product of mn can be obtained by more than 1 ways, some of them are listed below:

• mn × 1
• 1 × mn
• m × n
• n × m

For example: Consider the number of elements present in a matrix to be 12. Thus the order of a matrix can be either of the one listed below:

$$\begin{array}{l}12 \times 1\end{array}$$
, or
$$\begin{array}{l}1 \times 12\end{array}$$
, or
$$\begin{array}{l}6 \times 2\end{array}$$
, or
$$\begin{array}{l} 2 \times 6\end{array}$$
, or
$$\begin{array}{l}4 \times 3\end{array}$$
, or
$$\begin{array}{l}3 \times 4\end{array}$$
.

Thus, we have 6 different ways to write the order of a matrix, for the given number of elements.

Let us now look at a way to create a matrix for a given funciton:

For

$$\begin{array}{l} P_{ij} = i-2j \end{array}$$
, let us construct a 3 × 2  matrix.
So, this matrix will have 6 elements as following:

$$\begin{array}{l} P =\left[ \begin{matrix} P_{11} & P_{12}\cr P_{21} & P_{22} \cr P_{31} & P_{32} \cr \end{matrix} \right] \end{array}$$

Now, we will calculate the values of the elements one by one. To calculate the value of

$$\begin{array}{l} p_{11} \end{array}$$
, substitute
$$\begin{array}{l} i = 1 \space and \space j=1 \space in \space p_{ij} = i – 2j \end{array}$$
.

$$\begin{array}{l} P_{11} = 1 – (2 × 1) = -1 \end{array}$$

$$\begin{array}{l} P_{12} = 1 – (2 × 2) = -3 \end{array}$$

$$\begin{array}{l} P_{21} = 2 – (2 × 1) = 0 \end{array}$$

$$\begin{array}{l} P_{22} = 2 – (2 × 2) = -2 \end{array}$$

$$\begin{array}{l} P_{31} = 3 – (2 × 1) = 1 \end{array}$$

$$\begin{array}{l} P_{32} = 3 – (2 × 2) = -1 \end{array}$$

Hence,

$$\begin{array}{l} P =\left[ \begin{matrix} -1 & -3\cr 0 & -2 \cr 1 & -1 \cr \end{matrix} \right]_{3 × 2} \end{array}$$

There you go! You now know what order of matrix is, and how to determine it. To know more, download BYJU’S-The Learning App and study in an innovative way.