Order of Matrices

An order of matrices gives an idea of how many rows and columns are present in the matrix. When we define a matrix, it is the array of elements that are generally arranged in rows and columns. The order of matrices helps to get the numbers of rows and columns. The order helps understand the type of matrix and the total elements present in the matrix.

The order of matrices plays an important role that helps to understand if a particular operation of arithmetics can be performed for the two matrices.

In this article, we will understand the order of matrices and how the matrix order can be determined. 

Order of Matrix/Matrices

The order or dimension of matrices provides the dimension of the matrices and gives us the number of rows and columns in the matrix.

Generally, the matrix order is represented via Amxn, where m represents the number of rows in the matrix and n represents the number of columns in the matrix.

Also, when we carry out the multiplication, the product of the matrix order (m x n) provides the number of different elements in the matrix.

Let us go through the following example.

Columns

A mxn=

In the matrix, we observe m number of rows and n number of columns. The first number in the matrix order generally represents the number of rows and the number positioned second in the order represents the number of columns in the matrix.

Determining the Matrix Order

In order to understand the matrix order, let us go through the following example:

A=

B=

The above two matrices are A and B. 

Generally, the notation of any matrix is represented by

A= [aij]mxn where 1≤ i ≤ m, 1≤j ≤ n and i, j ∈ N

Through this, you can understand that the matrix is being denoted with an uppercase letter, while the elements are being denoted in a lower case with the same element. aij represents the matrix element, which is in the ith row and jth column.

So, if we take the case of bij, it represents an element of matrix B. Now let us go through the following example:

A=

We can say the element a21 represents the element in the 1st column and the 2nd row of the matrix A.

Hence, we can say a21 = 12. Hence b32=9 and b13 = 13.

So, if we take the matrix

B=

The notation for 2 would be b11 as it is in the 1st row and 1st column.

If there is a matrix with n columns and m rows, it is appropriate to say that the matrix is of the order m x n. 

With our earlier example, A is a 2 x 3 matrix, while B is a 4 x 3 matrix.

A = 2 X 3

B= 4 X 3

If you want to find out the order of any matrix, count the number of rows and columns. This way you will easily get the matrix order.

However, note the following points.

1. The order of a matrix is represented via a x b, and the number of elements in a matrix would be equal to the product of a and b.

2. The order of matrices shares an essential relationship with the no. of elements within a matrix.

Number of Elements in a Matrix

In the above-mentioned examples, A is the order of the matrix 2 x 3. Hence the number of elements there within a matrix is the product of 2 and 3, which is 6.

Also, if we take the other matrix B, it is of the order 4 x3 where the product of 4 x3 is 112, and hence 12 elements would be there.

It gives us an essential conclusion that when we know the order of a particular matrix, easily we can quickly determine the number of elements.

Hence, we can say that the m x n order matrix has mn elements.

If the Number of Elements is mn, then the order is m x n?

Suppose the number of elements is mn and the order is m x n. It is not true as the product total ‘mn’ can be got through various ways like

1. m x n

2. mn x 1

3.  n x m

4. 1 x mn

Now let us go through the following example. Suppose in a matrix the number of elements is 12. It can be in the order of the following

1. 12 x 1

2. 1 x 12

3. 6 x 2

4. 2 x 6

5. 4 x 3

6. 3 x 4 

So now we have six ways to write the order of a matrix.

Hence the number of elements if mn then the order is not necessarily m x n.

Create a Matrix for a Function

Pij = i – 2j so let us make a 3 x 2 matrix.

Hence the matrix will have the following 6 elements

P=

Now let us calculate each of the values of elements 

So for P11 i= 1 and j=1 hence Pij = I – 2j 

So P11 = 1 – (2×1) = -1

P12 = 1 – (2×2) = -3

P21 = 2 – (2×1) = 0

P22= 2 – (2×2) = -2

P31 = 3 – (2×1) = 1

P32 = 3 – (2×2) = -1

Hence

P=

So, this is how you create a matrix from a function.