Common Matrix Types and Their Key Characteristics
Before discussing the types of matrix, let's discuss what a matrix is.
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 matrix.
The size of a matrix is referred to as ‘ n by m′ matrix and is written as \[m \times n\], 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.
\[ \begin{bmatrix} 2 & 5 & 6 \\ 5 & 2 & 7 \end{bmatrix} \] known as a \[2 \times 3 \] matrix.
What are the Different Types of Matrices?
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) Skew -symmetric matrix
10) Horizontal matrix
11) Vertical matrix
12) Identity matrix
(Image will be uploaded soon)
Let's discuss the different types of matrices in mathematics, types of matrix in detail, matrix definition and types.
1. What is a Null Matrix?
If in a matrix all the elements are zero then it is called a zero matrix and it is generally denoted by 0. Thus,
A=\[ \left[a_{ij} \right] m \times n \] is a zero-matrix if \[a_{ij}\]=0 for all i and j
The first matrix O is a 2×2 matrix with all the elements equal to zero and the second matrix O is a 3×3 matrix with all the elements equal to zero.
\[ O = \begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix}, O = \begin{bmatrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{bmatrix} \]
2. What is a Triangular Matrix?
A square matrix is known to be triangular if all of its elements above the principal diagonal are zero then the triangular matrix is known as a lower triangular matrix or all of its elements below the principal diagonal are zero then the triangular matrix is known as upper triangular matrix.
A square matrix is known to be triangular if all of its elements above the principal diagonal are zero then the triangular matrix is known as a lower triangular matrix or all of its elements below the principal diagonal are zero then the triangular matrix is known as upper triangular matrix.
\[ \begin{bmatrix} 1 & 2 & 3 \\ 0 & 6 & 5 \\ 0 & 0 & 9 \end{bmatrix} \]
The matrix given above is a 3×3 upper triangular matrix.
The matrix given below is an example of a 3×3 lower triangular matrix.
\[ \begin{bmatrix} 1 & 0 & 0 \\ 2 & 4 & 0 \\ 3 & 5 & 6 \end{bmatrix} \]
3. What is a Vertical Matrix?
A matrix of order m×n is known as a vertical matrix of m>n, where m is equal to the number of rows and n is equal to the number of columns.
Matrix Example
\[ \begin{bmatrix} 2 & 5 \\ 1 & 1 \\ 3 & 6 \\ 2 & 4 \end{bmatrix} \]
In the matrix example given below the number of rows (m)=4, whereas the number of columns (n)=2. Therefore, this makes the matrix a vertical matrix.
4. What is a Horizontal Matrix?
A matrix of order m×n is known as a horizontal matrix if n>m, where m is equal to the number of rows and n is equal to the number of columns.
Matrix Example
\[ \begin{bmatrix} 1 & 2 & 3 & 4 \\ 2 & 5 & 1 & 1 \end{bmatrix} \]
In the matrix example given below the number of rows (m) = 2, whereas the number of columns (n) = 4. Therefore, we can say that the matrix is a horizontal matrix.
5. What is a Row Matrix?
A matrix that has only one row is known as a row matrix. Thus A = aijm×n
is a row matrix if m is equal to 1.
1. It is known so because it has only one row and the order of a row matrix will hence always be equal to \[1 \times n\].
Example of a Row matrix,
\[ A= \begin{bmatrix} 4 & 6 & 9\end{bmatrix}, B = \begin{bmatrix} 7 & 2 & 1 & 9 & 2 & 5 \end{bmatrix} \]
In the matrix example given above, matrix A has only one row and so matrix B has one row, therefore both matrices A and B are row matrices.
6. What is a Column Matrix?
A matrix that has one column is known as a Column matrix. Thus A = aij m×n is a column matrix if n is equal to 1.
1. It is known so because it has only one column and the order of a column matrix will hence always be equal to \[m \times 1\].
Example of a Column matrix,
\[ A = \begin{bmatrix} 3 \\ 4 \\ 8 \end{bmatrix}, B = \begin{bmatrix} 4 \\ 9 \\ 8 \\ 2 \end{bmatrix} \]
In the matrix example given above, matrix A has only one column and matrix B has one column, therefore both matrices A and B are column matrices.
7. What is a Diagonal Matrix?
If all the elements of the matrix, except the principal diagonal in any given square matrix, is equal to zero, it is known as a diagonal matrix. Thus a square matrix A=\[ \left[a_{ij}\right] \] is a diagonal matrix if \[a_{ij}= 0 \], when i is not equal to j
For Example,
\[ \begin{bmatrix} 2 & 0 & 0 \\ 0 & 3 & 0 \\ 0 & 0 & 4 \end{bmatrix} \]
The example given above is a diagonal matrix as it has elements only in its diagonal.
8. What is a Symmetric Matrix?
A square matrix A=\[ \left[a_{ij}\right] \] is known as a Symmetric matrix if \[a_{ij}=a_{ji}\], for all i,j values.
For Example,
\[ \begin{bmatrix} 1 & 2 & 3 \\ 2 & 4 & 5 \\ 3 & 5 & 2 \end{bmatrix} \]
9. What is the Skew -Symmetric Matrix?
A square matrix A=\[ \left[a_{ij}\right] \] is a skew-symmetric matrix if \[a_{ij}=a_{ji}\], for all values of i,j. Thus, in a skew-symmetric matrix all diagonal elements are equal to zero.
For Example,
\[ \begin{bmatrix} 0 & 2 & 1 \\ -2 & 0 & -3 \\ -1 & 3 & 0 \end{bmatrix} \]
10. What is an Identity Matrix?
If all the elements of a principal diagonal in a diagonal matrix are 1 , then it is called a unit matrix. A unit matrix of order n can be denoted by In. Thus, a square matrix A [aij]m×n is an identity matrix if all its diagonals have value 1.
For Example,
\[A = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix} \]
Questions to be Solved
1. Give an example of an identity matrix with a number of rows and columns equal to two.
Ans: We know that an identity matrix is one with its diagonal elements equal to 1 and all other elements equal to zero.
For example,
\[A = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} \]
How Do Students Prepare Notes on Matrices?
Read from the page that’s available on Vedantu- Types of Matrices
Understand the concepts and then write them down in your own words
Go through each of the solved questions
Make a note of the repeated questions or the similar questions
Highlight all the formulas in some colour
Go through the FAQs and then take note of the stuff that’s pertinent
Make a note of all the explanatory remarks
Revise from your book prior to exams
Importance of Matrices
Matrices are yet again an interesting chapter of Maths. A matrix is usually a rectangular array of numbers or of symbols that are arranged in rows as well as columns. The different types of matrices such as Row matrix, Column matrix, Null matrix, Square matrix, Diagonal matrix, Upper triangular matrix, Lower triangular matrix, Symmetric matrix, Skew -symmetric matrix, Horizontal matrix, Vertical matrix and Identity matrix have been described with the help of examples.
FAQs on Types of Matrices Explained with Examples
1. What are the main types of matrices students learn in the CBSE Class 12 curriculum?
In the CBSE Class 12 Maths syllabus for the 2025-26 session, students primarily focus on the following types of matrices:
- Row Matrix: A matrix with only one row.
- Column Matrix: A matrix with only one column.
- Square Matrix: A matrix where the number of rows equals the number of columns.
- Diagonal Matrix: A square matrix where all non-diagonal elements are zero.
- Scalar Matrix: A diagonal matrix where all diagonal elements are equal.
- Identity (or Unit) Matrix: A scalar matrix where all diagonal elements are 1.
- Zero (or Null) Matrix: A matrix where all elements are zero.
- Symmetric Matrix: A square matrix where the transpose is equal to the original matrix (A' = A).
- Skew-Symmetric Matrix: A square matrix where the transpose is equal to the negative of the original matrix (A' = -A).
2. What is the fundamental difference between a symmetric and a skew-symmetric matrix?
The fundamental difference lies in their relationship with their transpose. A square matrix 'A' is symmetric if it is equal to its transpose (i.e., aij = aji for all i, j). In contrast, a square matrix 'A' is skew-symmetric if it is equal to the negative of its transpose (i.e., aij = -aji for all i, j). A key property of a skew-symmetric matrix is that all its principal diagonal elements must be zero.
3. What defines a square matrix, and how are diagonal, scalar, and identity matrices related to it?
A square matrix is a matrix that has an equal number of rows and columns (order n x n). The other three types are special cases of a square matrix:
- Diagonal Matrix: This is a square matrix where all the elements outside the main diagonal are zero. The diagonal elements can be any value.
- Scalar Matrix: This is a specific type of diagonal matrix where all the elements on the main diagonal are identical (e.g., all 5s).
- Identity Matrix: This is a specific type of scalar matrix where all the main diagonal elements are 1. It is the multiplicative identity in matrix algebra.
Therefore, every identity matrix is a scalar matrix, and every scalar matrix is a diagonal matrix, and every diagonal matrix is a square matrix.
4. Why is the identity matrix so important in matrix operations?
The identity matrix, denoted by 'I', is crucial because it acts as the multiplicative identity in matrix algebra, similar to the number '1' in regular arithmetic. When any square matrix 'A' is multiplied by the identity matrix 'I' of the same order, the result is the original matrix 'A' (i.e., A * I = I * A = A). This property is fundamental for solving systems of linear equations and is essential for finding the inverse of a matrix.
5. Can a matrix be both a row matrix and a column matrix simultaneously? Explain why or why not.
Yes, a matrix can be both a row matrix and a column matrix at the same time. This occurs when the matrix has only one row and one column. Such a matrix is a 1x1 square matrix. For example, the matrix [7] has one row (making it a row matrix) and one column (making it a column matrix). It contains a single element and satisfies the conditions for both types.
6. How does the concept of a 'zero matrix' differ from the number zero in regular algebra?
While both serve as additive identities, they differ in key ways. The number zero is a single scalar value. A zero matrix, or null matrix, is a matrix of a specific size (m x n) where every element is zero. Unlike the number zero, a zero matrix has dimensions. Furthermore, in regular algebra, if xy = 0, then either x=0 or y=0. In matrix algebra, the product of two non-zero matrices can result in a zero matrix (i.e., AB = O does not necessarily mean A=O or B=O).
7. What are some real-world examples where different types of matrices are used?
Matrices are widely used in various fields:
- Computer Graphics: Square matrices are used for 3D transformations like rotation, scaling, and translation of objects.
- Cryptography: Matrices are used to encrypt and decrypt sensitive data, making information secure.
- Economics: Input-output models use matrices to represent the flow of goods and services between different sectors of an economy.
- Physics and Engineering: Matrices are used to solve systems of linear equations in circuit analysis, quantum mechanics, and optics.
8. What is the distinction between a horizontal matrix and a vertical matrix?
The distinction is based on the comparison between the number of rows and columns. A matrix is called a horizontal matrix if the number of columns is greater than the number of rows (n > m). Conversely, a matrix is called a vertical matrix if the number of rows is greater than the number of columns (m > n). A square matrix is neither horizontal nor vertical.
