Types of Matrices
There are so many different types of matrices in linear algebra. All types of matrices are differentiated based on their elements, order, and certain set of conditions. Matrices is the plural form of a matrix and matrixes is the less commonly used word to denote matrices.
In this article, let's learn about some of the commonly used types of matrices, their definition with examples.
Table of Contents
What Are the Different Types of Matrices?
This article describes some of the important types of matrices that are used in mathematics, engineering, and science. Here is the list of the most commonly used types of matrices in linear algebra:
 Row Matrix
 Column Matrix
 Singleton Matrix
 Rectangular Matrix
 Square Matrix
 Identity Matrices
 Matrix of ones
 Zero Matrix
 Diagonal Matrix
We can use these different types of matrices to organize data by age group, person, company, month, and so on. We can then use this information to make decisions and solve a lot of math problems.
Identifying Types of Matrices Based on Dimension
Matrices are in all sorts of sizes, but usually, their shapes remain the same. The size of a matrix is called its dimension which is the total number of rows and columns in a given matrix. In the belowgiven image, we can see how the dimension of a matrix is calculated.
In this section, let's learn to identify the types of matrices based on their dimension:
Row and Column Matrix
Matrices with only one row and any number of columns are known as row matrices and matrices with one column and any number of rows are called column matrices. Let's look at two examples below:
Row Matrix  Column Matrix 

\(A=\left[\begin{array}{ll} 1 & 0 & 2 & 4\\ \end{array}\right]\) 
\(B=\left[\begin{array}{c} 3 \\ 2 \\ 5 \end{array}\right]\) 
There is only one row, so A is a row matrix  There is only one column, so B is a column matrix 
Rectangular and Square Matrix
Any matrix that does not have an equal number of rows and columns is called a rectangular matrix and a rectangular matrix can be denoted by \([\mathrm{B}]_{\mathrm{m} \times \mathrm{n}}\). Any matrix that has an equal number of rows and columns is called a square matrix and a square matrix can be denoted by \([\mathrm{B}]_{\mathrm{n} \times \mathrm{n}}\). Let's look at the examples below:
Rectangular Matrix  Square Matrix 

\(B = \left[\begin{array}{ccc} 2 & 1 & 3 & 5 \\ 0 & 5 & 2 & 7\\ 1 & 1 & 2 & 9 \end{array}\right] \) 
\(C = \left[\begin{array}{ccc} 2 & 1 & 3 \\ 0 & 5 & 2 \\ 1 & 1 & 2 \end{array}\right] \) 
There are three rows and four columns in this matrix, so B is a rectangular matrix  There are three rows and three columns in this matrix, so C is a square matrix 
Constant Matrices
Constant matrices are matrices in which all the elements are constants for any given dimension/size of the matrix. The matrix elements are denoted by \(b_{ij}\). Let's look at these types of matrices whose elements are always constant.
Identity Matrix  Matrix of Ones  Zero Matrix 

The identity matrix is a square diagonal matrix, in which all entries on the main diagonal are equal to 1, and the rest of the elements are equal to 0. It is denoted by I.  Any matrix in which all the elements are equal to 1 is called a matrix of ones.  Any matrix in which all the elements are equal to 0 is called a zero matrix. 
\(I = \left[\begin{array}{ccc} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}\right] \) 
\(C = \left[\begin{array}{ccc} 1 & 1 & 1 \\ 1 & 1 & 1 \\ 1 & 1 & 1 \end{array}\right] \) 
\(D = \left[\begin{array}{ccc} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array}\right] \) 
Other Types of Matrices
Apart from the most commonly used matrices, there are other types of matrices that are used in advanced mathematics and computer technologies. Following are some of the other types of matrices:
Singular and Nonsingular Matrix
Any square matrix whose determinant is equal to 0 is called a singular matrix and any matrix whose determinant is not equal to 0 is called a nonsingular matrix. Determinant of a matrix can be found by using determinant formula. Let's look at two examples below:
Singular Matrix  Nonsingular Matrix 

C = \(\left[\begin{array}{ccc} 1 & 1 & 1 \\ 1 & 1 & 1 \\ 1 & 1 & 1 \end{array}\right] \) 
D = \(\left[\begin{array}{ccc} 2 & 1 & 1 \\ 1 & 2 & 1 \\ 1 & 1 & 1 \end{array}\right] \) 
C = \(\begin{vmatrix} 1 & 1 & 1 \\ 1 & 1 & 1 \\ 1 & 1 & 1 \end{vmatrix}\) 
D = \(\left\begin{array}{ccc} 2 & 1 & 1 \\ 1 & 2 & 1 \\ 1 & 1 & 1 \end{array}\right \) 
C = \(1 \times\left\begin{array}{ll} 1 & 1 \\ 1 & 1 \end{array}\right1 \times\left\begin{array}{ll} 1 & 1 \\ 1 & 1 \end{array}\right+1 \times\left\begin{array}{ll} 1 & 1 \\ 1 & 1 \end{array}\right\) 
D = \(2 \times\left\begin{array}{lll} 2 & 1 \\ 1 & 1 \end{array}\right1 \times\left\begin{array}{ll} 1 & 1 \\ 1 & 1 \end{array}\right+1 \times\left\begin{array}{ll} 1 & 2 \\ 1 & 1 \end{array}\right\) 
=1×(1×11×1)1×(1×11×1)+1×(1×11×1) =1×(11)1×(11)+1×(11) =1×(0)1×(0)+1×(0) =0+0+0 =0 
=2×(2×11×1)1×(1×11×1)+1×(1×12×1) =2×(21)1×(11)+1×(12) =2×(1)1×(0)+1×(1) =201 =1 
Here, C = 0, so C is a singular matrix  Here, D ≠ 0, so D is a nonsingular matrix 
Diagonal Matrix
A square matrix in which all the elements are 0 except for those elements that are in the diagonal is called a diagonal matrix. Let's take a look at the examples of different kinds of diagonal matrices: A scalar matrix is a special type of square diagonal matrix, where all the diagonal elements are equal.
Diagonal Matrix  Scalar Matrix 

B = \(\left[\begin{array}{llll} 
C = \(\left[\begin{array}{lll} 3 & 0 & 0 \\ 0 & 3 & 0 \\ 0 & 0 & 3 \end{array}\right]\) 
Here, we can see that except for the diagonal elements, all the other elements are zero. Hence, B is a diagonal matrix.  Here, we can see that the diagonal elements are equal and all the other elements are zero. Hence, C is a scalar matrix. 
Upper and Lower Triangular Matrix
An upper triangular matrix is a square matrix where all the elements that are present below the diagonal elements are 0. A lower triangular matrix is a square matrix where all the elements that are present above the diagonal elements are 0. Let's look at the examples below:
Upper Triangular Matrix  Lower Triangular Matrix 

B = \(\left[\begin{array}{lll} 3 & 2 & 1 \\ 0 & 4 & 5 \\ 0 & 0 & 6 \end{array}\right]\) 
C = \(\left[\begin{array}{lll} 3 & 0 & 0 \\ 4 & 1 & 0 \\ 2 & 7 & 9 \end{array}\right]\) 
Here, we can see that all the elements that are present below the leftright diagonal are 0. Hence, B is an upper triangular matrix. 
Here, we can see that all the elements that are present above the leftright diagonal are 0. Hence, B is an upper triangular matrix. 
Symmetric and Skew Symmetric Matrix
A square matrix D of size n×n is considered to be symmetric if and only if D^{T}= D. A square matrix F of size n×n is considered to be skewsymmetric if and only if F^{T}=  F. Let's consider the examples of two matrices D and F:
Symmetric Matrix  Skewsymmetric Matrix 

D = \(\left[\begin{array}{lll} 
F = \(\left[\begin{array}{cc} F^{T} = \(\left[\begin{array}{cc} F = \(\left[\begin{array}{cc} 
Here, D = D^{T}. Hence, D is a symmetric matric.  Here, F^{T} = F, Hence F is a skewsymmetric matrix. 
Boolean Matrix
A matrix is considered to be a boolean matrix when all its elements are either 1s and 0s. Let's consider the example of the matrix B to understand this better:
B = \(\left[\begin{array}{lll}
0 & 1 & 0 \\
1 & 0 & 1 \\
0 & 1 & 1
\end{array}\right]\)
Stochastic Matrices
A square matrix C is considered to be stochastic when all of its entries are nonnegative and when the entries in each column sum to 1. Consider the example of the matrix C here:
C = \(\left[\begin{array}{lll}
0.3 & 0.4 & 0.5 \\
0.3 & 0.4 & 0.3 \\
0.4 & 0.2 & 0.2
\end{array}\right]\)
Orthogonal Matrix
A square matrix B is considered to be an orthogonal matrix, when B × B^{T} = I, where I is an identity matrix and B^{T} is the transpose of matrix B. Take an example of the matrix B:
B =\(\left[\begin{array}{ll} 0 & 1 \\ 1 & 0 \end{array}\right]\) 
B^{T }=\(\left[\begin{array}{ll} 0 & 1 \\ 1 & 0 \end{array}\right]\) 
B × B^{T} = \(\left[\begin{array}{ll} = \(\left[\begin{array}{ll} = \(\left[\begin{array}{ll} Here, we can see that B × B^{T} = I. Hence, B is an orthogonal matrix 
Related Articles on Types of Matrices
Check out the following pages related to types of matrices
 Matrix Calculator
 Matrix formula
 How to Solve Matrices
 Diagonal Matrix Calculator
 Transpose Matrix Calculator
Important Notes on Types of Matrices
Here is a list of a few points that should be remembered while studying the types of matrices
 Matrices with only one row and any number of columns are known as row matrices
 Matrices with one column and any number of rows are called column matrices
 Constant matrices are matrices in which all the elements are constants for any given dimension/size of the matrix
Examples on Types of Matrices

Example 1: Which of the following is not a type of matrix?
a) Square matrix
b) Diagonal matrix
c) Row matrix
d) Minor matrixSolution: d.) Minor matrix cannot be considered as a type of matrix. Square, diagonal, and row matrices are various types of matrices.

Example 2: Identify the type of the matrices A and B:
\(A=\left[\begin{array}{ll}
0 & 0 \\
0 & 0
\end{array}\right]\)\(B=\left[\begin{array}{ll}
1 & 0 \\
0 & 2
\end{array}\right]\)
Solution:Matrix A can be considered as a null or a zero matrix since all the elements in this matrix are 0. Matrix B is a diagonal matrix since it's a square matrix where all elements except the diagonal elements are zero.
FAQs on Types of Matrices
How Do You Identify Types of Matrices?
One of the ways to identify the type of a matrix is by checking its dimension. The dimension of a matrix is the total number of rows and columns in a given matrix. Consider the example of matrix B = [ 1 2 5 7 0]. In this matrix, there is one row and 5 columns, and hence its dimension is 1 × 5. If a matrix has one row and many columns, then it can be considered as a row matrix, and hence the matrix B is a row matrix.
What Are the Different Types of Matrices?
Here is the list of the different types of matrices:
 Row & column matrix
 Singular & nonsingular matrix
 Rectangular & square matrix
 Identity matrix
 Matrix of ones
 Zero matrix
 Diagonal matrix
 Upper & lower triangular matrix
What Type of Matrix Is a 2x2?
A square matrix is a matrix with the dimension n × n, i.e, it has an equal number of rows and columns. A 2 × 2 type of matrix has 2 rows and 2 columns and hence it can be considered as a square matrix.
Which Type of Matrix Is Never Invertible?
A square matrix is said to be invertible if and only if its determinant is not equal to zero. For example, a 2 x 2 matrix is only invertible if the determinant of this matrix is not 0. If the determinant of this matrix is 0, then the matrix is not invertible and it cannot have a reverse. Thus, any singular matrix whose determinant is equal to zero is never invertible.
(a b c) Is Which Type of Matrix?
(a b c) is a matrix with one row and three columns and this type of matrix with one row and many columns is called a row matrix. Thus, (A B C) is a row matrix.
What Types of Matrices Have Inverses?
Matrices whose determinant is not equal to zero can have inverses. A nonsingular matrice is the type of matrix whose determinant is not equal to zero, and hence we can find the inverse for a nonsingular matrix.
What Is a Triangular Matrix and Its Types?
A triangular matrix is considered to be a special type of square matrix in which either all the entries above the diagonal elements or all the entries below the diagonal elements are zero. The two types of a triangular matrix are the upper triangular matrix and the lower triangular matrix. An upper triangular matrix is a square matrix where all the elements that are present below the diagonal elements are 0. A lower triangular matrix is a square matrix where all the elements that are present above the diagonal elements are 0.
Which Type of Matrix Is Called a Null Matrix?
Zero Matrix is the type of matrix that is called a null matrix since all the elements in a zero matrix are equal to zero.