 # Determine The Order of Matrix

### Order of a Matrix

Before we know what the order of a matrix means, let’s first understand what matrices are. Matrices can be defined as a rectangular array of numbers or functions. Since a matrix is a rectangular array, it is 2-dimensional. A two-dimensional matrix basically consists of the number of rows which is denoted by (m) and a number of columns denoted by (n). Let us understand the concept in a better way with some examples.

### What is a Matrix?

• A matrix is a rectangular array of numbers or symbols which are generally arranged in rows and columns.

• The order of the matrix is defined as the number of rows and columns.

• The entries are the numbers in the matrix and each number is known as an element.

• The plural of matrix is matrices.

• The size of a matrix is referred to as ‘n by m’ matrix and is written as m×n, where n is the number of rows and m is the number of columns.

• For example, we have a 3×2 matrix, that’s because the number of rows here is equal to 3 and the number of columns is equal to 2.

### What are the Different Types of Matrix?

There are different types of matrices. Here they are –

1) Row matrix

2) Column matrix

3) Null matrix

4) Square matrix

5) Diagonal matrix

6) Upper triangular matrix

7) Lower triangular matrix

8) Symmetric matrix

9) Anti-symmetric matrix

### How Will You Determine the Order of a Matrix?

If we have a matrix that has m number of rows and n number of columns, then let’s know how to find the order of matrix.

Here are a few examples how to find order of matrix,

$\begin{bmatrix} 15 & 9 & -5 \end{bmatrix}$

The order of the above matrix is (1×3), since the number of rows (m) = 1 and the number of columns (n) = 3.

$\begin{bmatrix} 7 & -6 \end{bmatrix}$

The order of the above matrix is (1×2), since the number of rows (m) = 1 and the number of columns (n) = 2.

$\begin{bmatrix} a & b\\ c& d \end{bmatrix}$

The order of the above matrix is (2×2), since the number of rows (m) = 2 and the number of columns (n) = 2.

$\begin{bmatrix} 8 & a & 5\\ -3& 15 & b \end{bmatrix}$

The order of the above matrix is (2×3), since the number of rows (m) = 2 and the number of columns (n) = 3.

We can clearly see, a matrix of the order m × n has mn elements. Hence, we can say that if the number of elements in a matrix be prime, then it must have one row or one column.

Usually, we denote a matrix by using a capital letter, such as A, B, C, D, M, N, X, Y, Z, etc.

A Small NOTE!

• It is quite fascinating that there is a relation between the number of elements present in a matrix and the order of the matrix.

• The order of a matrix is denoted by m × n, and the number of elements present in a matrix will always be equal to the product of m and n.

$\begin{bmatrix} 8 & a & 5\\ -3& 15 & b \end{bmatrix}$

In the example given above, what is the order of a matrix? The matrix order math is 2 × 3. Therefore, the number of elements present in the above matrix will also be 2 times 3, that is equal to 6. Now, when we count the number of elements in the matrix, it adds up to 6.

$\begin{bmatrix} 7 & -6 \end{bmatrix}$

Similarly, what is the order of a matrix? The matrix order math is 1 × 2, thus the number of elements present will be 1 times 2 that is equal to 2. Now, when we count the number of elements in the matrix, it adds up to 2.

This gives us an important insight that if we know the order of matrix, it would be easy for us to determine the total number of elements present in the matrix. The conclusion,

If the order of matrix is m × n, it will have mn (product of m and n) elements.

Now , you might wonder whether the converse of the previous statement is true?

The converse of the previous statement says that: If the number of element is equal to mn, so the order would be m × n. But, this is definitely not true. This is because the product of mn can be obtained by more than one ways; some of the ways are listed below:

1.  mn × 1

2. 1 × mn

3. m × n

4. n × m

### Questions to Be Solved:

Question 1) If a matrix A has six number of elements, then determine the order of the matrix.

Solution) We know that the number of elements is 6.Now you might think what is the order of a matrix? Let’s write down all the possible factors of the number 6.

6 = 1 × 6

6 = 6 × 1

6 = 2 × 3

6 = 3 × 2

We can get the number 6 in the following 4 ways.

Therefore, there are four possible orders of the matrix with 6 number of elements, that is , 6 = 1 × 6, 6 × 1, 2 × 3 and 3 × 2.

Question 2) What is the order of a matrix given below?

A =   $\begin{bmatrix} 3 & 4 & 9\\ 12& 11 & 35 \end{bmatrix}$

Solution) The number of rows in the above matrix A = 2

The number of columns in the above matrix A= 3 .

Therefore, the order of the matrix is 2 × 3.

1. What is Matrix and Types?

A matrix can be defined as a rectangular array of numbers. A matrix consists of rows and columns (m×n). There are different types of matrices.

Here they are-

1) Row matrix

2) Column matrix

3) Null matrix

4) Square matrix

5) Diagonal matrix

6) Upper triangular matrix

7) Lower triangular matrix

8) Symmetric matrix

9) Anti-symmetric matrix

2. What is the Order of a Square Matrix?

A matrix which has the number of rows equal to the number of columns, is known as a square matrix. In this matrix, the elements are arranged in m number of rows and n number of columns and the order of matrix is denoted as m×n.

3. Can the Rank of a Matrix Be Zero?

Yes, the rank of a matrix can be zero. But this happens only in the case of a zero matrix. The Rank of a matrix is equal to the number of non-zero rows in the row echelon form (REF). Since in a zero matrix, there is no non-zero row, the rank is equal to 0.

4. What is Rank of Matrix With an Example?

The rank of a matrix can be defined as the maximum number of linearly independent rows in a any matrix A is known as the row rank of A , and the maximum number of linearly independent columns in matrix A is known as the column rank of A.