Types of Matrices: Definition, Types, and Solved Examples

Matrices are rectangular arrays or tables arranged in rows and columns of numbers, symbols, or expressions. Based on the number of rows and columns, we can define the order of the matrices. There are several types of matrices based on the rows, columns, and elements of the matrices, such as square matrix, row matrix, column matrix, diagonal matrix, null matrix, scalar matrix, identity matrix, triangular matrix, symmetric and skew-symmetric matrices, and so on.

Matrices form an important part of mathematics and the concepts are helpful in higher studies as well. Hence, students are advised to read it thoroughly. In this article, we have discussed different types of matrices, their operations, examples, and their properties. Read on to find more!

Definition of Matrix and Their Types

Matrices are one of the most significant tools in mathematics. Arthur Cayley is known as the “Father of Matrices”.

Matrices are rectangular arrays or tables arranged in rows and columns of numbers, symbols, or expressions. The matrices are denoted by a capital letter \((e . g ., A, B, X)\). The matrix elements are represented by lowercase letters with a double subscript \(\left(\right.\) e.g.: \(\left.a_{i j}, b_{i j}, x_{i j}\right)\).

Mathematically, the matrix is denoted by \(A = {\left[ {{a_{ij}}} \right]_{m \times n}}\), where \({a_{ij}}\) belongs to \(i^{t h}\) row and \(j^{t h}\) column.

A general representation of matrix with an order \(m \times n\) is given by \([A]_{m \times n}\), where

1. \(m\) – Number of rows in matrix \(A\).
2. \(n\) – Number of columns in matrix \(A\).

Types of Matrices

There are different types of matrices that we use. They are discussed below:

Row Matrix

Row matrix is the matrix having only one row in it. The order of the row matrix is \(\left[a_{i j}\right]_{1 \times n}\) Where \(n-\) is the number of columns. The general representation of the row matrix is given by

\(\left[\begin{array}{lllll}a_{11} & a_{12} & a_{13} & \ldots \ldots & a_{1 n}\end{array}\right]\)

Examples:
\(\left[\begin{array}{lll}2 & 3 & 5\end{array}\right],\left[\begin{array}{lll}5 & 8\end{array}\right]\)

Column Matrix

Column matrix is the matrix having only one column in it. The order of the column matrix is \(\left[a_{i j}\right]_{m \times 1}\) Where \(m-\) is the number of rows. The general representation of the column matrix is given by

\(\left[ \begin{array}{l} {a_{11}}\\ {a_{21}}\\ {a_{31}}\\ .\\ .\\ {a_{n1}} \end{array} \right]\)

Examples:
\(\left[ \begin{array}{l} 1\\ 2\\ 3 \end{array} \right],\,\left[ \begin{array}{l} a\\ b \end{array} \right]\)

Null or Zero Matrix

A matrix in which all the elements are zero is known as a zero matrix or null matrix. Thus, matrix \(A = {\left[{{a_{ij}}} \right]_{m \times n}}\) is said to be a null matrix if \({a_{ij}} = 0\) for all the values of \(i, j\).

\({\left[ {\begin{array}{*{20}{c}} 0&0\\ 0&0 \end{array}} \right]_{2 \times 2}},\,{\left[ {\begin{array}{*{20}{c}} \begin{array}{l} 0\\ 0\\ 0 \end{array}&\begin{array}{l} 0\\ 0\\ 0 \end{array} \end{array}} \right]_{3 \times 2}}\,,\,{\left[ {\begin{array}{*{20}{c}} 0&0&0\\ 0&0&0\\ 0&0&0 \end{array}} \right]_{3 \times 3}}\)

Vertical Matrix

A matrix, in which the number of rows is more than the number of columns, is called the vertical matrix. Thus, matrix \(A=\left[a_{i j}\right]_{m \times n}\) is said to be a vertical matrix, in which \(m>n\). The vertical matrix has fewer columns.

Example:
\({\left[ {\begin{array}{*{20}{c}} {{a_{11}}}&{{a_{12}}}\\ \begin{array}{l} {a_{21}}\\ {a_{31}} \end{array}&\begin{array}{l} {a_{22}}\\ {a_{32}} \end{array} \end{array}} \right]_{3 \times 2}}\)

Horizontal Matrix

A matrix in which the number of rows is less than the number of columns is called the horizontal matrix. Thus, matrix \(A=\left[a_{i j}\right]_{m \times n}\) is said to be a horizontal matrix, in which \(m<n\). The horizontal matrix has more columns.

Example:
\({\left[ {\begin{array}{*{20}{c}} \begin{array}{l} {a_{11}}\\ {a_{21}} \end{array}&\begin{array}{l} {a_{12}}\\ {a_{22}} \end{array}&\begin{array}{l} {a_{13}}\\ {a_{23}} \end{array} \end{array}} \right]_{2 \times 3}}\)

Rectangular Matrix

A matrix that has an unequal number of rows and columns is known as a rectangular matrix. Thus, matrix \(A=\left[a_{i j}\right]_{m \times n}\) is said to be a rectangular matrix if \(m \neq n\).

Example:
\(\left[ {\begin{array}{*{20}{c}} \begin{array}{l} {a_{11}}\\ {a_{21}} \end{array}&\begin{array}{l} {a_{12}}\\ {a_{22}} \end{array}&\begin{array}{l} {a_{13}}\\ {a_{23}} \end{array} \end{array}} \right]\)

Square Matrix

A matrix that has an equal number of rows and columns is known as a square matrix. The order of the square matrix is generally \(2 \times 2,3 \times 3,4 \times 4\) and so on. Thus, matrix \(A=\left[a_{i j}\right]_{m \times n}\) is said to be a square matrix if \(m=n\).

The square matrix with an order of \(2 \times 2\) is given by

\(\left[ {\begin{array}{*{20}{c}} {{a_{11}}}&{{a_{12}}}\\ {{a_{21}}}&{{a_{22}}} \end{array}} \right]\)

The square matrix with an order of \(3 \times 3\) is given by

\(\left[ {\begin{array}{*{20}{c}} \begin{array}{l} {a_{11}}\\ {a_{21}}\\ {a_{31}} \end{array}&\begin{array}{l} {a_{12}}\\ {a_{22}}\\ {a_{32}} \end{array}&\begin{array}{l} {a_{13}}\\ {a_{23}}\\ {a_{33}} \end{array} \end{array}} \right]\)

The square matrix has many applications, such as these are used to find the determinants.

Diagonal Matrix

A square matrix, in which all non-diagonal elements are zeros, is known as a diagonal matrix. In a diagonal matrix, the diagonal elements have non-zero values, and the non-diagonal elements are zeroes. Thus, a square matrix \(A = {\left[{{a_{ij}}} \right]_{m \times m}}\) is said to be a diagonal matrix if \({a_{ij}} = 0\) for \(i \neq j\)

Example:

\(\left[ {\begin{array}{*{20}{c}} {{a_{11}}}&0\\ 0&{{a_{22}}} \end{array}} \right],\,\left[ {\begin{array}{*{20}{c}} \begin{array}{l} {a_{11}}\\ 0\\ 0 \end{array}&\begin{array}{l} 0\\ {a_{22}}\\ 0 \end{array}&\begin{array}{l} 0\\ 0\\ {a_{33}} \end{array} \end{array}} \right]\)

Scalar Matrix

A square matrix, in which all non-diagonal elements are zeros and all diagonal elements are equal, is known as the scalar matrix. The scalar matrix is the diagonal matrix, in which elements of the principal diagonal are equal to the same constant value.

Thus, a square matrix \(A = {\left[{{a_{ij}}} \right]_{m \times m}}\) is said to be a diagonal matrix if \({a_{ij}} = 0\) for \(i \neq j\) and \(a_{i j}=k\) (some constant) for \(i=j\).

Example:

\(\left[ {\begin{array}{*{20}{c}} a&0\\ 0&a \end{array}} \right],\,\left[ {\begin{array}{*{20}{c}} \begin{array}{l} 5\\ 0\\ 0 \end{array}&\begin{array}{l} 0\\ 5\\ 0 \end{array}&\begin{array}{l} 0\\ 0\\ 5 \end{array} \end{array}} \right]\)

Identity Matrix or Unit Matrix

A square matrix, in which all non-diagonal elements are zeros all diagonal elements are equal to one, is known as an identity matrix or unit matrix. A diagonal matrix, in which elements of the principal diagonal are equal to the one \((1)\) known as the unit matrix.

Thus, a square matrix \(A = {\left[{{a_{ij}}} \right]_{m \times m}}\) is said to be a diagonal matrix if \({a_{ij}} = 0\) for \(i \neq j\) and \(a_{i j}=1\) for \(i=j\)

Generally, the unit matrix or identity matrix of order \(n\), is denoted by ” \(I_{n}\) “. Thus, the determinant of the identity matrix is one.

Examples:

\(\left[ {\begin{array}{*{20}{c}} 1&0\\ 0&1 \end{array}} \right],\,\left[ {\begin{array}{*{20}{c}} \begin{array}{l} 1\\ 0\\ 0 \end{array}&\begin{array}{l} 0\\ 1\\ 0 \end{array}&\begin{array}{l} 0\\ 0\\ 1 \end{array} \end{array}} \right]\)

Triangular Matrices

A square matrix in which the elements above or below the principal diagonal are zeros is known as a triangular matrix. There are two types of triangular matrices.

1. Upper triangular matrix
2. Lower triangular matrix

Upper Triangular Matrix

The upper triangular matrix is the square matrix, in which all the elements below the principal diagonal are zero. Thus, a square matrix \(A = {\left[{{a_{ij}}} \right]_{m \times m}}\) is said to be an upper triangular matrix if \({a_{ij}} = 0\) for \(i>j\).

Examples:
The upper triangular matrix with an order of \(3 \times 3\) is given below:

\(\left[ {\begin{array}{*{20}{c}} \begin{array}{l} 1\\ 0\\ 0 \end{array}&\begin{array}{l} 2\\ 4\\ 0 \end{array}&\begin{array}{l} 3\\ 5\\ 6 \end{array} \end{array}} \right]\)

Lower Triangular Matrix

The lower triangular matrix is the square matrix, in which all the elements above the principal diagonal are zero. Thus, a square matrix \(A = {\left[{{a_{ij}}} \right]_{m \times m}}\) is said to be a lower triangular matrix if \({a_{ij}} = 0\) for \(i<j\).

Example:
The lower triangular matrix with an order of \(3 \times 3\) is given below:

\(\left[ {\begin{array}{*{20}{c}} \begin{array}{l} 1\\ 2\\ 3 \end{array}&\begin{array}{l} 0\\ 4\\ 5 \end{array}&\begin{array}{l} 0\\ 0\\ 6 \end{array} \end{array}} \right]\)

Singleton Matrix

A matrix that has only one element is called a singleton matrix. The order of the singleton matrix is \(1×1\).

Example:

\([a]_{1 \times 1}\)

Singular Matrix

A square matrix of any order is said to be singular if the determinant of the square matrix is zero.

The determinant of the matrix \(A\) of order two, \(A = \left[ {\begin{array}{*{20}{c}} {{a_{11}}}&{{a_{12}}}\\ {{a_{21}}}&{{a_{22}}} \end{array}} \right],\) is calculated as follows:

\(\det \,A = \left| {\begin{array}{*{20}{c}} {{a_{11}}}&{{a_{12}}}\\ {{a_{21}}}&{{a_{22}}} \end{array}} \right| = {a_{11}}{a_{22}} – {a_{12}}{a_{21}}\)

The determinant of the matrix \(A\) of order two, \(A = \left[ {\begin{array}{*{20}{c}} {{a_{11}}}&{{a_{12}}}&{{a_{13}}}\\ {{a_{21}}}&{{a_{22}}}&{{a_{23}}}\\ {{a_{31}}}&{{a_{32}}}&{{a_{33}}} \end{array}} \right]\) is calculated as follows:

\(\left| A \right| = \det \,A = \left[ {\begin{array}{*{20}{c}} {{a_{11}}}&{{a_{12}}}&{{a_{13}}}\\ {{a_{21}}}&{{a_{22}}}&{{a_{23}}}\\ {{a_{31}}}&{{a_{32}}}&{{a_{33}}} \end{array}} \right] = {a_{11}}\left| {\begin{array}{*{20}{c}} {{a_{22}}}&{{a_{23}}}\\ {{a_{32}}}&{{a_{33}}} \end{array}} \right| – {a_{12}}\left| {\begin{array}{*{20}{c}} {{a_{21}}}&{{a_{23}}}\\ {{a_{31}}}&{{a_{33}}} \end{array}} \right| + {a_{13}}\left| {\begin{array}{*{20}{c}} {{a_{21}}}&{{a_{22}}}\\ {{a_{31}}}&{{a_{32}}} \end{array}} \right|\)

Symmetric Matrix

A square matrix is said to be symmetric if its transpose matrix equals the given matrix. In a symmetric matrix, matching elements on either side of the principal diagonal are the same. Thus, a square matrix \(A = {\left[{{a_{ij}}} \right]_{m \times m}}\) is said to be a symmetric matrix if \(a_{i j}=a_{j i}\).

\(A=A^{T}\)

Example:
Let \(A = \left[ {\begin{array}{*{20}{c}} 1&2&3\\ 2&4&5\\ 3&5&6 \end{array}} \right]\) and its transpose is \({A^T} = \left[ {\begin{array}{*{20}{c}} 1&2&3\\ 2&4&5\\ 3&5&6 \end{array}} \right].\)

Skew-Symmetric Matrix

A square matrix, in which all the diagonal elements are zeroes and negative of the transpose matrix equals the given matrix. Thus, a square matrix \(A = {\left[{{a_{ij}}} \right]_{m \times m}}\) is said to be a skew-symmetric matrix if \(a_{i j}=-a_{j i}\) for all \(i \neq j\) and \(a_{i j}=0\) for all \(i=j\).

\(A=-A^{T}\)

Example:
Let \(A = \left[ {\begin{array}{*{20}{c}} 0&2&3\\ { – 2}&0&{ – 5}\\ { – 3}&5&0 \end{array}} \right]\) and \({A^T} = \left[ {\begin{array}{*{20}{c}} 0&{ – 2}&{ – 3}\\ 2&0&5\\ 3&{ – 5}&0 \end{array}} \right] = – \left[ {\begin{array}{*{20}{c}} 0&2&3\\ { – 2}&0&{ – 5}\\ { – 3}&5&0 \end{array}} \right] = \,-A\)

Hermitian Matrix

A square matrix is a Hermitian matrix if its complex conjugate and the given matrix are the same. Thus, a square matrix \(A = {\left[{{a_{ij}}} \right]_{m \times m}}\) is said to be a hermitial matrix if \(a_{i j}=\bar{a}_{j i}\) for all \(i \neq j\) and \({a_{ij}} = \) real for all \(i=j\).

Example:

\(\left[ {\begin{array}{*{20}{c}} 3&{2 + 3\,i}&{ – 3 + i}\\ {2 – 3\,i}&2&{ – 5 + 4\,i}\\ { – 3 – i}&{ – 5 – 4\,i}&5 \end{array}} \right]\)

Special Types of Matrices

There are some special types of matrices based on the index or power of the matrices. They are

Idempotent Matrix

A square matrix \(A\) of any order is said to be idempotent; if \(A^{2}=A\).

Nilpotent Matrix

A square matrix of an index \(n, n \in N\), is said to be nilpotent if \(A^{n}=0\) and \(A^{n-1} \neq 0\).

Here, \(A\) is said to be nilpotent of index \(n\).

Involutary Matrix

A square matrix \(A\) of any order is said to be involuntary if \(A^{2}=I\). the matrix, which equals its inverse, is known as the involuntary matrix.

\(A=A^{-1}\)

Periodic Matrix

A square matrix of any order satisfies the condition \(A^{p+1}=A\), for some positive integer \(p\), then matrix \(A\) is said to be periodic with period \(p\).

A periodic matrix with period one is known as an idempotent matrix.

Solved Examples

Q.1. Check whether the matrix \(\left[ {\begin{array}{*{20}{c}} 1&0&0\\ 0&4&0\\ 0&0&6 \end{array}} \right]\) is a diagonal matrix or not?
Ans: In a diagonal matrix, the diagonal elements are non-zero values, and the non-diagonal elements are zeroes. Thus, a square matrix \(A = {\left[{{a_{ij}}} \right]_{m \times m}}\) is said to be a diagonal matrix if \(a_{i j}=0\) for \(i \neq j\).
Given matrix is \(\left[ {\begin{array}{*{20}{c}} 1&0&0\\ 0&4&0\\ 0&0&6 \end{array}} \right]\)
Here, \(a_{i j}=0\) for \(i \neq j\). So, the given matrix is a diagonal matrix.

Q.2. For the given square matrix of order \(3 \times 3,\,\left[ {\begin{array}{*{20}{c}} {{a_{11}}}&{{a_{12}}}&{{a_{13}}}\\ {{a_{21}}}&{{a_{22}}}&{{a_{23}}}\\ {{a_{31}}}&{{a_{32}}}&{{a_{33}}} \end{array}} \right],\) write the upper triangular matrix and lower triangular matrix.
Ans: Given matrix is \(\left[ {\begin{array}{*{20}{c}} {{a_{11}}}&{{a_{12}}}&{{a_{13}}}\\ {{a_{21}}}&{{a_{22}}}&{{a_{23}}}\\ {{a_{31}}}&{{a_{32}}}&{{a_{33}}} \end{array}} \right]\)
The upper triangular matrix is the square matrix, in which all the elements below the principal diagonal are zero. Thus, a square matrix \(A = {\left[{{a_{ij}}} \right]_{m \times m}}\) is said to be an upper triangular matrix if \(a_{i j}=0\) for \(i>j\).
Upper triangular matrix \(\left[ {\begin{array}{*{20}{c}} {{a_{11}}}&{{a_{12}}}&{{a_{13}}}\\ 0&{{a_{22}}}&{{a_{23}}}\\ 0&0&{{a_{33}}} \end{array}} \right]\)
The lower triangular matrix is the square matrix, in which all the elements above the principal diagonal are zero. Thus, a square matrix \(A = {\left[{{a_{ij}}} \right]_{m \times m}}\) is said to be a lower triangular matrix if \(a_{i j}=0\) for \(i<j\).
Lower triangular matrix \(\left[ {\begin{array}{*{20}{c}} {{a_{11}}}&0&0\\ {{a_{21}}}&{{a_{22}}}&0\\ {{a_{31}}}&{{a_{32}}}&{{a_{33}}} \end{array}} \right]\)

Q.3. Write the identity matrix of order \(3\).
Ans: A square matrix, in which all non-diagonal elements are zeros all diagonal elements are equal to one, is known as an identity matrix or unit matrix. A diagonal matrix, in which elements of the principal diagonal are equal to the one \(1\) known as the unit matrix.
So, the identity matrix of order \(3\) is \(\left[ {\begin{array}{*{20}{c}} 1&0&0\\ 0&1&0\\ 0&0&1 \end{array}} \right].\)

Q.4. Check whether the matrix \(\left[ {\begin{array}{*{20}{c}} 1&2\\ 3&4 \end{array}} \right]\) is a singular matrix or not?
Ans: Let \(A = \left[ {\begin{array}{*{20}{c}} 1&2\\ 3&4 \end{array}} \right]\)
Determinant of a given matrix, \(\det \,A = \left| {\begin{array}{*{20}{c}} 1&2\\ 3&4 \end{array}} \right|\)
\(A=4 \times 1-2 \times 3\)
\(A=4-6=-2 \neq 0\)
So, the given matrix is a non-singular matrix.

Q.5. Check whether matrix \(A = \left[ {\begin{array}{*{20}{c}} 1&2\\ 2&4 \end{array}} \right]\) is symmetric or skew-symmetric.
Ans: Given \(A = \left[ {\begin{array}{*{20}{c}} 1&2\\ 2&4 \end{array}} \right]\)
Transpose of matrix \(A\) is changing rows to columns and columns to rows.
\({A^T} = \left[ {\begin{array}{*{20}{c}} 1&2\\ 2&4 \end{array}} \right]\)
Here, \(A = {A^T} = \left[ {\begin{array}{*{20}{c}} 1&2\\ 2&4 \end{array}} \right]\)
A square matrix is said to be symmetric if its transpose matrix equals the given matrix.
So, the given matrix is symmetric.

Summary

In this article, we studied the definition of the matrix, which is the array of rectangular arrangements of rows and columns. Here, we learned the order of the matrix, which is given by the number of rows × number of columns. 

In this article, we discussed the various types of the matrices such as row matrix, column matrix, rectangular matrix, square matrix, diagonal matrix, scalar matrix, null or zero matrices, unit or identity matrix, upper triangular matrix, lower triangular matrix, singleton matrix, singular matrix, symmetric and skew-symmetric matrices. We also learned some special types of matrices such as idempotent, periodic, nilpotent, and involuntary matrices.

FAQs

Q.1. What is a column matrix?
Ans: Column matrix is the matrix having only one column in it.

Q.2. What is the \(2 \times 3\) matrix called?
Ans: The given matrix has different rows and columns. We know that a matrix in which unequal rows and columns are called a rectangular matrix. So, the \(2 \times 3\) matrix is called a rectangular matrix.

Q.3. What is a singleton matrix?
Ans: A matrix that has only one element is called a singleton matrix.

Q.4. What is the difference between a diagonal matrix and a scalar matrix?
Ans: In a diagonal matrix, the diagonal elements are non-zero values, and the non-diagonal elements are zeroes. In the scalar matrix, diagonal elements are the same.

Q.5. What are the types of triangular matrices?
Ans: There are two types of matrices. They are
1. Upper triangular matrix
2. Lower triangular matrix

We hope you find this detailed article on types of matrices helpful. If you have any doubts or queries, feel to ask us in the comment section. Happy learning!