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.

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 & Skew-Hermitian 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.

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 below-given 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:

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:

Identity and Zero Matrices

Let's look at the identity matrix and zero matrix.

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 Non-singular 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 non-singular matrix. The determinant of a matrix can be found by using determinant formula. Let's look at two examples below:

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.

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:

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 skew-symmetric if and only if F^T= - F. Let's consider the examples of two matrices D and F:

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: \(\left[\begin{array}{cc}
  3 & 3-2 i \\ \\
  3+2 i & 2
  \end{array}\right]\)
• 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 non-negative and when the entries in each column sum to 1. Similarly, a matrix with all its entries as non-negative 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:

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 Aⁿ = A, for every n ≥ 2. For example, A² = A, A³ = A, and so on. To check whether a square matrix A is idempotent, it is sufficient to check whether A² = 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 A² = 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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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
• Non-singular matrix
• Rectangular matrix
• Square matrix
• Identity matrix
• Matrix of ones
• Zero matrix
• Diagonal matrix
• Upper matrix
• Lower triangular matrix
• Orthogonal matrix
• Symmetric matrix
• Skew-symmetric matrix
• Hermitian matrix
• Skew-Hermitian 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 non-singular matrix is the type of matrix whose determinant is not equal to zero, and hence we can find the inverse for a non-singular 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.