Definition and properties of different types of matrices with examples
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 in Linear Algebra Explained
1. What are the different types of matrices in mathematics?
The main types of matrices in mathematics include row, column, square, rectangular, zero, identity, diagonal, scalar, triangular, symmetric, and skew-symmetric matrices. These are classified based on their order and element arrangement.
- Row matrix: Only one row
- Column matrix: Only one column
- Square matrix: Same number of rows and columns
- Zero matrix: All elements are 0
- Identity matrix: Diagonal elements are 1, others 0
- Diagonal matrix: Non-diagonal elements are 0
- Scalar matrix: Diagonal elements are equal
- Triangular matrix: Elements above or below diagonal are 0
- Symmetric matrix: A = AT
- Skew-symmetric matrix: AT = −A
2. What is a square matrix?
A square matrix is a matrix with the same number of rows and columns, i.e., of order n × n. For example, [[1, 2], [3, 4]] is a 2 × 2 square matrix. Square matrices are important because only they have determinants, inverses, and eigenvalues.
3. What is the difference between a row matrix and a column matrix?
The difference is that a row matrix has only one row, while a column matrix has only one column.
- Row matrix: Order 1 × n, example: [1 2 3]
- Column matrix: Order m × 1, example: [[1], [2], [3]]
4. What is a zero matrix?
A zero matrix is a matrix in which all elements are equal to 0. It is denoted by O and can be of any order m × n. Example:
- 2 × 2 zero matrix = [[0, 0], [0, 0]]
5. What is an identity matrix?
An identity matrix is a square matrix with 1s on the main diagonal and 0s elsewhere. It is denoted by In. Example (2 × 2):
- I2 = [[1, 0], [0, 1]]
6. What is a diagonal matrix?
A diagonal matrix is a square matrix in which all non-diagonal elements are 0. Example:
- [[3, 0, 0], [0, 5, 0], [0, 0, 7]]
7. What is a scalar matrix?
A scalar matrix is a diagonal matrix in which all diagonal elements are equal. Example:
- [[4, 0], [0, 4]]
8. What is a symmetric matrix?
A symmetric matrix is a square matrix that is equal to its transpose, meaning A = AT. Example:
- [[1, 2], [2, 3]]
9. What is a skew-symmetric matrix?
A skew-symmetric matrix is a square matrix that satisfies AT = −A. This means diagonal elements are always 0. Example:
- [[0, 2], [-2, 0]]
10. What is a triangular matrix?
A triangular matrix is a square matrix in which either all elements above or below the main diagonal are zero.
- Upper triangular matrix: Elements below diagonal are 0
- Lower triangular matrix: Elements above diagonal are 0
