Order of a Matrix Definition Formula and Solved Examples
Before we know what the order of a matrix means, let’s first understand what matrices are. Matrices can be defined as rectangular arrays of numbers or functions. Since a matrix is a rectangular array, it is 2-dimensional. A two-dimensional matrix 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 the matrix is matrices.
The size of a matrix is referred to as the ‘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.
How Will You Determine the Order of a Matrix?
The order of a matrix can easily be determined by counting the number of rows and columns the matrix consists of. 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 the matrix.
Here are a few examples of how to find the order of a matrix,
[159−5]
[15]
The order of the above matrix is (1×3), since the number of rows (m) = 1 and the number of columns (n) = 3.
[7−6]
[7]
The order of the above matrix is (1×2) since the number of rows (m) = 1 and the number of columns (n) = 2.
[abcd]
[ac]
The order of the above matrix is (2×2) since the number of rows (m) = 2 and the number of columns (n) = 2.
[8a5−315b]
[8−3]
The order of the above matrix is (2×3) since the number of rows (m) = 2 and the number of columns (n) = 3.
The order of the matrix below is 3 x 4, which means that it has 3 rows and 4 columns.
[173242466913]
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 is prime, then it must have one row or one column. For determining the order of a matrix of square matrices like 1 x 1, 2 x 2, 3 x 3,……., n x n the order will be represented by the number of rows or number of columns that is n.
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.
Example: [8a5−315b]
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, which is equal to 6.
This gives us an important insight that if we know the order of the matrix, it would be easy for us to determine the total number of elements present in the matrix. In conclusion, if the order of the 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 elements is equal to mn, then the order would be m × n. But, this is definitely not true. This is because the product of mn can be obtained in more than one way; some of the ways are listed below:
mn × 1
1 × mn
m × n
n × m
What are the Different Types of Matrix?
There are different types of matrices. Here they are –
Row Matrix: In this matrix, the number of rows is 1 and this is fixed while the number of columns may vary.
Example: [137]1×3[1]
1×3
Column Matrix: A type of matrix that contains only one column and any no of rows is known as a column matrix.
Example: [1234]4×1
4×1
Singleton Matrix: This type of matrix has the number of rows and columns the same that is 1 which means there will be only one element in the matrix.
Example: [5]1×1[5]
1×1
Rectangular Matrix: A rectangular matrix is a type of matrix that has a different number of rows and columns. A rectangular matrix is defined as
A
m×n.
Example: The below example is showing a 3x4 matrix.
[173242466913]
Square Matrix: A square matrix is a type of matrix that consists of the same number of rows and columns. The representation of the square matrix is
A
n×n.
Example: The below example is showing a 3x3 square matrix.[832646579]
Null Matrix: A type of matrix having all elements as 0 is known as a null matrix. Example: [000000000]
Diagonal Matrix: A type of matrix that has all elements as zero except diagonal elements is known as a diagonal matrix.
Example: [800040009]
Scalar Matrix: Scalar matrix is a type of matrix in which the diagonal value is the same and all the rest values are zero. So, it is a kind of diagonal matrix where all diagonal elements are the same.
Example: [400040004]
Identity Matrix: The identity matrix is a type of scalar matrix having the diagonal value 1 and all the rest values are 0. The identity matrix always has an equal number of rows and columns.
Example: [100010001]
Upper Triangular Matrix: Upper Triangular Matrix is a type of matrix in which the triangular elements above the diagonal are non-zero and triangular elements below the diagonal are zero.
Example: [832046009]
Lower Triangular Matrix: Lower Triangular Matrix is a type of matrix in which the triangular elements above the diagonal are zero and triangular elements below the diagonal are non-zero.
Example: [800640579]
Symmetric Matrix: Symmetric matrix is a type of matrix which has values equal to its transpose A= AT, i.e., amn = anm . The square matrix is the type of symmetric matrix.
Example: [123245356]
Anti-symmetric Matrix: Anti-symmetric matrix is a type of matrix which has negative values to its transpose A= - AT , i.e., amn = -anm. This type of matrix is also called a “skew-symmetric matrix”
Example: [0−2−32053−50]
Solved Examples
Question 1) If a matrix A has six numbers 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 numbers 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= [349121113]
[312]
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.
FAQs on How to Determine the Order of a Matrix
1. What is the order of a matrix?
The order of a matrix is defined as the number of rows and columns it contains, written in the form m × n. Here, m represents the number of rows and n represents the number of columns. For example, if a matrix has 2 rows and 3 columns, its order is 2 × 3. The order helps determine the size and structure of a matrix in linear algebra.
2. How do you determine the order of a matrix?
To determine the order of a matrix, count the number of rows and columns in it. Follow these steps:
- Count the horizontal lines to get the number of rows (m).
- Count the vertical lines to get the number of columns (n).
- Write the order as m × n.
For example, if a matrix has 3 rows and 2 columns, its order is 3 × 2.
3. What is the order of a 3x4 matrix?
The order of a 3 × 4 matrix is 3 rows and 4 columns. This means the matrix has 3 horizontal lines of elements and 4 elements in each row. In general, a matrix written as m × n always has m rows and n columns.
4. What is the difference between rows and columns in a matrix?
In a matrix, rows are horizontal arrangements of elements, while columns are vertical arrangements of elements. Specifically:
- Rows (m): Counted from left to right.
- Columns (n): Counted from top to bottom.
The order of a matrix depends on both rows and columns and is written as m × n.
5. What is the order of a square matrix?
A square matrix has the same number of rows and columns, so its order is n × n. For example:
- A 2 × 2 matrix is a square matrix.
- A 3 × 3 matrix is also a square matrix.
Square matrices are important in determinants, inverses, and eigenvalues in linear algebra.
6. What is the order of a row matrix?
A row matrix has only one row, so its order is 1 × n. Here, n represents the number of columns. For example, a matrix with elements [2 5 7] has order 1 × 3. Row matrices are also called row vectors.
7. What is the order of a column matrix?
A column matrix has only one column, so its order is m × 1. Here, m represents the number of rows. For example, a matrix with elements arranged vertically as 4, 6, 9 has order 3 × 1. Column matrices are also known as column vectors.
8. Can two matrices be added if their orders are different?
Two matrices can be added only if they have the same order. Matrix addition requires that both matrices have equal numbers of rows and columns. For example:
- A matrix of order 2 × 3 can be added only to another matrix of order 2 × 3.
- Matrices of order 2 × 3 and 3 × 2 cannot be added.
This rule ensures that corresponding elements can be added correctly.
9. What is the order of the product of two matrices?
The order of the product of two matrices is determined by the outer dimensions and is m × p if the matrices are of order m × n and n × p. Matrix multiplication is possible only when the number of columns in the first matrix equals the number of rows in the second matrix. For example:
- If A is 2 × 3 and B is 3 × 4, then AB is 2 × 4.
10. Why is the order of a matrix important?
The order of a matrix is important because it determines whether matrix operations like addition, subtraction, and multiplication are possible. Specifically:
- Addition and subtraction require matrices of the same order.
- Multiplication depends on matching inner dimensions.
- Square matrices (n × n) are required for determinants and inverses.
Thus, knowing the matrix order is essential for solving problems in linear algebra and coordinate geometry.
