Types of Matrices

Introduction

A matrix that consists of several objects arrayed in a rectangular order. The word "matrix" is a plural matrix and is not commonly used to refer to a matrix. There are many types of matrices, which are basically categorized by element value, order, number of rows and columns, and so on. In this tutorial, we will discuss types of matrices.

What is a Matrix?

A matrix is an arrangement of numbers, variables, symbols, or expressions in a rectangular table that contains different numbers of rows and columns. These are rectangular arrays that define various operations such as addition, multiplication, and transpose. The numbers or entries in the matrix are called their elements.

If the matrix has m rows and n columns, there are m × n elements. The matrix is represented by uppercase letters (in this case, "A"), and the elements in the matrix are represented by lowercase letters and two subscripts that represent the positions of the elements in the same order and in the same order. In the case of, '𝑎𝑖𝑗', where i is the number of rows and j is the number of columns.

What are the Different Types of Matrices?

There are many types of matrices, which are basically categorized based on element values, order, number of rows and columns, and so on. There are different types of matrices in linear algebra. All types of matrices are distinguished based on their elements, order, and specific conditions. Special types of matrices are square, diagonal, identity, translocations, and symmetric matrices. This is a square matrix with the same number of rows and columns. Currently, using different terms, different matrix types are categorized below, along with their definitions and examples.

What is a Null Matrix?

If all specified elements in a matrix are 0, then the matrix is called a null matrix and is usually represented by zero.

Therefore, if $\mathrm{a_{ij}\:=\:0}$ for all i and j, then A = [aij] m × n is a zero matrix.

$$\mathrm{A\:=\begin{bmatrix} 0 & 0 \\ 0 & 0 \\ \end{bmatrix}}$$

What is a Triangular Matrix?

It is a rectangular matrix with all the 0 factors below and/or above the diagonal. There are two principal types of triangular matrices

A square matrix with all the capabilities above the principal diagonal 0 is referred to as the lower triangle matrix.

A rectangular matrix with all the functions below the principal diagonal zero is known as the top triangular matrix .

For example,

$$\mathrm{A\:=\begin{bmatrix} 1 & 2 & 3 \\ 0 & 3 & 5\\ 0 & 0 & 6 \end{bmatrix}}$$

The above matrix is an example of an upper triangular matrix.

For example,

$$\mathrm{B\:=\begin{bmatrix} 1 & 0 & 0 \\ 2 & 3 & 0\\ 4 & 5 & 6 \end{bmatrix}}$$

The above matrix is an example of a lower triangular matrix

What is a Vertical Matrix?

A vertical matrix is a matrix with more rows than columns.

For example,

$$\mathrm{C\:=\begin{bmatrix} 1 & 5 \\ 0 & 6 \\ 0 & 3 \end{bmatrix}}$$

In this example, the number of rows is 3, and the number of columns is 2. Hence it is a vertical matrix

What is a Horizontal Matrix?

A horizontal matrix is a matrix with fewer rows than columns.

For example

$$\mathrm{D\:=\begin{bmatrix} 1 & 2 & 5 \\ 0 & 3 & 6\\ \end{bmatrix}}$$

In this example, the number of rows is 2, and the number of columns is 3.Hence it is the horizontal matrix.

What is a Row Matrix?

Row matrices contain only one row.

For example,

$$\mathrm{R\:=\begin{bmatrix} 1 & 2 & 3 \\ \end{bmatrix}}$$

Since only one row is present in the above matrix, it is called a row matrix

What is a Column Matrix?

The column matrix has only one column since there is only one column so the order of the column matrix will be given as always mx1

For example,

Since only one column is present in the above matrix, it is called column matrix

$$\mathrm{T\:=\begin{bmatrix} 1 \\ 0 \\ 2 \end{bmatrix}}$$

What is a Diagonal Matrix?

In linear algebra, a diagonal matrix is a matrix with all zero entries from the main diagonal. This term usually refers to a square matrix. The elements of the main diagonal can be either zero or nonzero.

For example,

$$\mathrm{A\:=\begin{bmatrix} 1 & 0 & 0 \\ 0 & 3 & 0\\ 0 & 0 & 6 \end{bmatrix}}$$

What is a Symmetric Matrix?

The matrix will be known as a symmetric matrix if the matrix is equal to its transpose.

$$\mathrm{M^{T}\:=\:M}$$

For example,

$$\mathrm{M\:=\begin{bmatrix} 1 & 2 & 3 \\ 2 & 4 & 5\\ 3 & 5 & 2 \end{bmatrix}}$$

What is the Skew-Symmetric Matrix?

A matrix will be known as a symmetric matrix if the matrix is equal to the negative of its transpose.

$$\mathrm{M^{T}\:=\:-M}$$

For example,

$$\mathrm{M\:=\begin{bmatrix} 0 & 2 & 4 \\ -2 & 0 & 3\\ -4 & -3 & 2 \end{bmatrix}}$$

What is an Identity Matrix?

If all diagonal elements of a matrix are one and the rest is zero, then that matrix will be known as the Identity matrix

For example,

$$\mathrm{M\:=\begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0\\ 0 & 0 & 1 \end{bmatrix}}$$

Since all the diagonal element is 1 and rest element s zero hence it is identity matrix or unit matrix.

Solved question

1)Identify the following matrices.

$$\mathrm{\begin{bmatrix} 1 & 0 & 0 \\ 0 & 5 & 0\\ 0 & 0 & 7 \end{bmatrix}}$$

Answer − It is a diagonal matrix since all only diagonal has non zero value.

2) Identify the following matrices.

$$\mathrm{\begin{bmatrix} 2 \\ 4 \\ 6 \end{bmatrix}}$$

Answer − Since in the above only one column is present hence it is called a column matrix

3) Identify the following matrices

$$\mathrm{\begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0\\ 0 & 0 & 1 \end{bmatrix}}$$

Answer − Since all the diagonal element is 1 and rest element s zero hence it is identity matrix or unit matrix.

4) Identify the type of matrices for the following matrices.

$$\mathrm{A\:=\:\begin{bmatrix} 1 & 5 & 2 \\ \end{bmatrix}}\:and\:B\:=\:\begin{bmatrix} 1 & 0 & 4 \\ \end{bmatrix}$$

Answer − Since in both cases matrices A and B both have only one row and three columns hence the given matrices are row matrices.

Conclusion

A matrix is usually a rectangular array of numbers or symbols arranged in both rows and columns. There are many types of matrices in linear algebra. All types of matrices are distinguished based on their elements, order, and specific conditions. Null Matrix, Triangular Matrix, Vertical Matrix, Horizontal, Diagonal Matrix, Symmetric, Skew- Symmetric, and an Identity Matrix

FAQs

1. What are the different types of matrices?

Null Matrix, Triangular Matrix, Vertical Matrix, Horizontal, Diagonal Matrix, Symmetric, Skew-Symmetric, and an Identity Matrix

2. What do you mean by Identity matrix also, give one example?

The diagonal element is 1 and the rest elements is zero hence it is an identity matrix or unit matrix. For example

$$\mathrm{\begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0\\ 0 & 0 & 1 \end{bmatrix}}$$

3. What do you mean by row matrix?

A matrix that has only one row is called a row matrix. Since there is only one row so the order of this matrix will be given as always 1xn

For example,

$$\mathrm{A\:=\:\begin{bmatrix} 1 & 5 & 2 \\ \end{bmatrix}}$$

4. What is the difference between a symmetric matrix and a skew-symmetric matrix?

A matrix will be known as a symmetric matrix if $\mathrm{M^{T}\:=\:M}$

A matrix will be known as a symmetric matrix if $\mathrm{M^{T}\:=\:-M}$

5. What is the difference between the upper triangular matrix and the lower triangular matrix?

It is a rectangular matrix with all the 0 factors below and/or above the diagonal. There are two principal types of triangular matrices.

A square matrix with all the capabilities above the principal diagonal 0 is referred to as the lower triangle matrix

A rectangular matrix with all the functions below the principal diagonal zero is known as the top (upper) triangular matrix.