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Types of Matrices
There are many different types of matrices in linear algebra. All types of matrices are differentiated based on their elements, order, and certain set of conditions. The word "matrices" is the plural form of a matrix and is the less commonly used to denote matrices.
In this article, let's learn about some of the commonly used types of matrices, and their definitions along with examples.
1.  What are Types of Matrices? 
2.  Identifying Types of Matrices 
3.  Other Types of Matrices 
4.  Special Matrices 
5.  FAQs on Types of Matrices 
What are 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
 Rectangular Matrix & Square Matrix
 Identity Matrix
 Zero Matrix
 Diagonal Matrix
 Singular Matrix & Nonsingular Matrix
 Hermitian Matrix & SkewHermitian Matrix
 Upper & Lower Triangular Matrices
 Symmetric Matrix and Skew Symmetric Matrix
 Orthogonal 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
Matrices are in all sorts of sizes, but usually, their shapes remain the same. The size of a matrix is called its order 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 B_{m × 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 B_{n × 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. 
Identity and Zero Matrices
Let's look at the identity matrix and zero matrix.
Identity Matrix  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 0 is called a zero matrix. 
\(I = \left[\begin{array}{ccc} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}\right] \) 
\(D = \left[\begin{array}{ccc} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array}\right] \) 
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. The 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 = \(\left\begin{array}{ccc} 2 & 1 & 1 \\ 1 & 2 & 1 \\ 1 & 1 & 1 \end{array}\right \) 
=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 main diagonal are 0. Hence, B is an upper triangular matrix. 
Here, we can see that all the elements that are present above the main 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. 
Hermitian and Skew Hermitian Matrices
There is a small difference between the symmetric and hermitian matrices.
 A matrix is said to be hermitian if and only it is equal to the transpose of its conjugate matrix.
Example: \(\left[\begin{array}{cc}
3 & 32 i \\ \\
3+2 i & 2
\end{array}\right]\)  A matrix is said to be skew hermitian if and only if it is equal to the negative of its conjugate matrix.
Example: \(\left[\begin{array}{cc}
3 i & 1+i \\
1+i & i
\end{array}\right]\)
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 stochastic matrix is a type of matrix whose all entries represent probability. A square matrix C is considered to be left stochastic when all of its entries are nonnegative and when the entries in each column sum to 1. Similarly, a matrix with all its entries as nonnegative such that entries in each row sum to 1 is called a right stochastic matrix. Consider the example of a left stochastic 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. 
Special Matrices
Apart from what we have learned so far, there are some special types of matrices:
Idempotent Matrix
A square matrix A is said to be an idempotent matrix if and only if A^{n} = A, for every n ≥ 2. For example, A^{2} = A, A^{3} = A, and so on. To check whether a square matrix A is idempotent, it is sufficient to check whether A^{2} = A.
Nilpotent Matrix
A square matrix A of order n is nilpotent if and only if A^{k} = O for some k ≤ n. For example, A = \(\left[\begin{array}{ll}
2 & 4 \\ \\
1 & 2
\end{array}\right]\) is a nilpotent matrix as A^{2} = O, where O is null matrix of order 2.
Involutory Matrix
A square matrix A is called an involutory matrix if and only A^{1} = A. For example, an identity matrix is involutory as it is equal to its inverse.
Important Notes on 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/order of the matrix.
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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 (which is made up of the minors of the corresponding elements) cannot be considered as a type of matrix. Square, diagonal, and row matrices are various types of matrices.
Answer: d) is not a type 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.
Answer: A is a zero matrix and B is a diagonal matrix.

Example 3: What type of matrix is A = \(\left[\begin{array}{ccc}
2 & 1 & 3 \\
0 & 5 & 2 \\
0 & 0 & 2
\end{array}\right]\)? Upper triangular or lower triangular?Solution:
In the given matrix, all the elements that are below the principal diagonal are zeros.
Hence, the matrix is upper triangular.
Answer: Upper triangular matrix
FAQs on Types of Matrices
Name the Different Types of Matrices.
Matrices are classified into several types based on their order, elements, and several other conditions. Here is the list of the different types of matrices:
 Row matrix
 Column matrix
 Singular matrix
 Nonsingular matrix
 Rectangular matrix
 Square matrix
 Identity matrix
 Matrix of ones
 Zero matrix
 Diagonal matrix
 Upper matrix
 Lower triangular matrix
 Orthogonal matrix
 Symmetric matrix
 Skewsymmetric matrix
 Hermitian matrix
 SkewHermitian matrix
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 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.
What Types of Matrices Have Inverses?
Matrices whose determinant is not equal to zero can have inverses. A nonsingular matrix is the type of matrix whose determinant is not equal to zero, and hence we can find the inverse for a nonsingular matrix.
(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 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 zeros. 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. It doesn't have an inverse as its determinant is 0 and hence it is a singular matrix.
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