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.

Order 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.

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

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

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

[latex] A = [a_{ij}]_{m × n} [/latex], where [latex] 1 ≤ i ≤ m , 1 ≤ j ≤ n [/latex] and [latex]i , j \in N [/latex]

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. [latex] a_{ij} [/latex] represents any element of matrix  which is in [latex] i^{th}[/latex]  row and [latex] j^{th} [/latex] column. Similarly,[latex] b_{ij} [/latex] represents any element of matrix B.

So, in the matrices given above, the element [latex] a_{21} [/latex]  represents the element which is in the [latex] 2^{nd} [/latex]row and the  [latex] 1^{st} [/latex] column of matrix A.

∴a21 = 12

Similarly, [latex] b_{32} = 9 , b_{13} = 13  [/latex] and so on.

Can you write the notation of 15 for matrix B ?

Since it is in [latex] 3^{rd} [/latex] row and 3rd column, it will be denoted by [latex] b_{33} [/latex].

If the matrix has [latex] m [/latex] rows and [latex] n [/latex] columns, it is said to be a matrix of the order [latex]m × n[/latex]. 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:

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

[latex] 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}
[/latex]

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:

[latex]12 \times 1[/latex], or [latex]1 \times 12[/latex], or [latex]6 \times 2[/latex], or [latex] 2 \times 6[/latex], or [latex]4 \times 3[/latex], or [latex]3 \times 4[/latex].

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 [latex] P_{ij} = i-2j [/latex] , let us construct a 3 × 2  matrix.
So, this matrix will have 6 elements as following:

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

Now, we will calculate the values of the elements one by one. To calculate the value of [latex] p_{11} [/latex] , substitute  [latex] i = 1 \space and \space j=1 \space in \space p_{ij} = i – 2j [/latex] .

[latex] P_{11} = 1 – (2 × 1) = -1 [/latex]

[latex] P_{12} = 1 – (2 × 2) = -3 [/latex]
[latex] P_{21} = 2 – (2 × 1) = 0 [/latex]
[latex] P_{22} = 2 – (2 × 2) = -2 [/latex]
[latex] P_{31} = 3 – (2 × 1) = 1 [/latex]
[latex] P_{32} = 3 – (2 × 2) = -1 [/latex]

Hence,
[latex] P =\left[
\begin{matrix}
-1 & -3\cr
0 & -2 \cr
1 & -1 \cr
\end{matrix}
\right]_{3 × 2}
[/latex]

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.

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