A matrix is a rectangular array of numbers. The numbers in a matrix are enclosed by [] or (). The size of a matrix is defined by the number of rows and columns in it. The matrices are classified into different types on the basis of the value of their element, size, number of rows, columns, etc. Learn the definitions, examples of matrices types in the following sections.
Classification of Matrices
The matrices are classified in various based depending on their size, order of a matrix, number of rows, no of columns, elements, the position of an element in matrix and others. Here the list of classification of matrices is provided.
- Row Matrix
- Column Matrix
- Zero or Null Matrix
- Singleton Matrix
- Horizontal Matrix
- Vertical Matrix
- Square Matrix
- Diagonal Matrix
- Scalar Matrix
- Identity (Unit) Matrix
- Equal Matrix
- Triangular Matrices
- Singular Matrix
- Non-Singular Matrix
- Symmetric Matrices
- Skew-Symmetric Matrices
- Hermitian Matrix
- Skew – Hermitian Matrix
- Orthogonal Matrix
- Idempotent Matrix
- Involuntary Matrix
- Nilpotent Matrix
Types of Matrices Based on Dimension
The detailed explanations of types of matrices based on their dimensioned are included here:
Row Matrix:
A matrix that has exactly only one row is called a row matrix.
Example:
\( A =\left[\begin{matrix}
8 & 5 & 2\cr
\end{matrix}
\right]
\)
Column Matrix:
A matrix that has only one column is called a column matrix.
Example:
\( B =\left[\begin{matrix}
3\cr
1\cr
7\cr
\end{matrix}
\right]
\)
Square Matrix:
Any matrix in the order of m x n and where m = n is called a square matrix or a matrix that has an equal number of rows and columns in it is a square matrix.
Example:
\( C =\left[\begin{matrix}
1 & 2 & 3\cr
9 & 8 & 7\cr
4 & 5 & 6\cr
\end{matrix}
\right]
\)
Rectangular Matrix:
A matrix of the order m x n and in which m ≠ n is called a rectangular matrix.
Example:
\( D =\left[\begin{matrix}
4 & 2 & 8\cr
6 & 1 & 5\cr
\end{matrix}
\right]
\)
Singleton Matrix:
A matrix that has only one element is called a singleton matrix. Its order is 1 x 1.
Examples:
\( M =\left[
\begin{matrix}
3 \cr
\end{matrix}
\right]
\), \( N =\left[
\begin{matrix}
12 \cr
\end{matrix}
\right]
\)
Horizontal Matrix:
A matrix in which the number of columns is more than the number of rows is called a horizontal matrix.
Example:
\( A =\left[\begin{matrix}
1 & 5 & 9\cr
13 & 18 & 25 \cr
\end{matrix}
\right]
\)
Vertical Matrix:
A matrix in which the number of rows is greater than the number of columns is called a vertical matrix.
Example:
\( C =\left[\begin{matrix}
2 & 4\cr
3 & 6\cr
4 & 8\cr
1 & 2\cr
\end{matrix}
\right]
\)
Diagonal Matrix:
A square matrix where all the elements, except the principal diagonal, are zero is called a diagonal matrix.
Example:
\( D =\left[\begin{matrix}
30 & 0 & 0\cr
0 & 6 & 0\cr
0 & 0 & 7\cr
\end{matrix}
\right]
\)
Scalar Matrix:
If all the elements in the diagonal of a diagonal matrix are equal, it is called a scalar matrix.
Example:
\( X =\left[
\begin{matrix}
5 & 0 & 0\cr
0 & 5 & 0\cr
0 & 0 & 5\cr
\end{matrix}
\right]
\), \( Y =\left[
\begin{matrix}
8 & 0\cr
0 & 8\cr
\end{matrix}
\right]
\)
Identity Matrix:
It is a square matrix in which all elements in the leading diagonal are 1 and the remaining elements are zeros. It is also called a unit matrix.
Examples:
\( E =\left[
\begin{matrix}
1 & 0 & 0\cr
0 & 1 & 0\cr
0 & 0 & 1\cr
\end{matrix}
\right]
\) and \( F =\left[
\begin{matrix}
1 & 0\cr
0 & 1\cr
\end{matrix}
\right]
\)
Matrix of Ones:
A matrix in which all the elements are 1, then it is called a matrix of ones.
Example:
\( G =\left[\begin{matrix}
1 & 1 & 1\cr
1 & 1 & 1\cr
1 & 1 & 1\cr
\end{matrix}
\right]
\)
Zero Matrix:
A matrix in which all the elements are zero is called a zero matrix or null matrix.
Example:
\( H =\left[\begin{matrix}
0 & 0 & 0\cr
0 & 0 & 0\cr
0 & 0 & 0\cr
\end{matrix}
\right]
\)
Explanations on Classification of Matrices
Triangular Matrix:
If the elements above or below the principal diagonal of a square matrix are zero, then it is a triangular matrix.
Examples:
\( D =\left[
\begin{matrix}
3 & 13 & 15\cr
0 & 6 & 8\cr
0 & 0 & 7\cr
\end{matrix}
\right]
\) and \( C =\left[
\begin{matrix}
8 & 0 & 0\cr
10 & 7 & 0\cr
11 & 15 & 4\cr
\end{matrix}
\right]
\)
Symmetric Matrix:
A square matrix is said to be symmetric matrix if the original matrix is equal to the transpose of the given matrix.
Example:
\( B =\left[
\begin{matrix}
8 & 10 & 6\cr
10 & 7 & 15\cr
6 & 15 & 4\cr
\end{matrix}
\right]
\)
and \( BT =\left[
\begin{matrix}
8 & 10 & 6\cr
10 & 7 & 15\cr
6 & 15 & 4\cr
\end{matrix}
\right]
\)
Here, B = BT;. So, these are symmetric matrices.
Skew-symmetric Matrix:
A square matrix is said to be skew-symmetric matrix if the original matrix is equal to the negative of the transpose of the given matrix.
Example:
\( F =\left[\begin{matrix}
0 & 3\cr
-3 & 0\cr
\end{matrix}
\right]
\) \( FT =\left[
\begin{matrix}
0 & -3\cr
3 & 0\cr
\end{matrix}
\right]
\) \( -F =\left[
\begin{matrix}
0 & -3\cr
3 & 0\cr
\end{matrix}
\right]
\)
So, FT = -F.
Questions on Types of Matrices
Question 1:
Identify the class of each matrix.
\( M =\left[
\begin{matrix}
3 & 4 & 9\cr
\end{matrix}
\right]
\), \( N =\left[
\begin{matrix}
3 & 4 & 9\cr
12 & 11 & 35 \cr
3 & 4 & 9\cr
\end{matrix}
\right]
\), \( O =\left[
\begin{matrix}
3 & 4\cr
12 & 11\cr
\end{matrix}
\right]
\)
Solution:
\( M =\left[
\begin{matrix}
3 & 4 & 9\cr
\end{matrix}
\right]
\)
M is a row matrix as it has exactly one row.
\( N =\left[
\begin{matrix}
3 & 4 & 9\cr
12 & 11 & 35 \cr
3 & 4 & 9\cr
\end{matrix}
\right]
\)
N is a square matrix as it has an equal number of rows and columns.
\( O =\left[
\begin{matrix}
3 & 4\cr
12 & 11\cr
\end{matrix}
\right]
\)
O is a square matrix. Because it has 2 rows, 2 columns.
Question 2:
Construct a null matrix of order 4 x 3 and unit matrix of order 4 x 4.
Solution:
A null matrix of order 4 x 3 is \( A =\left[
\begin{matrix}
0 & 0 & 0\cr
0 & 0 & 0\cr
0 & 0 & 0\cr
0 & 0 & 0\cr
\end{matrix}
\right]
\)
Identity matrix of order 4 x 4 is \( A =\left[
\begin{matrix}
1 & 0 & 0 & 0\cr
0 & 1 & 0 & 0\cr
0 & 0 & 1 & 0\cr
0 & 0 & 0 & 1\cr
\end{matrix}
\right]
\)
FAQ’s on Matrices Classification
1. What are the four types of the matrix?
The 4 different types of matrices are row matrix, column matrix, null matrix, and square matrix.
2. How many types of matrices are there?
On average, 22 types of matrices are there.
3. What are the properties of matrices?
The important properties of matrices are associative property, distributive property, identity property, property of zero, closure property and so on.
4. How to classify matrices?
Basically, matrices are classified as per the number of rows, number of columns, specific elements in them.