Types of Matrix: Overview, Questions, Preparation

You can check the different types of matrix tables with crisp and concise information. The detailed explanation for each type of matrix with identification, application, and solved examples is provided below.

Matrix type	Specification
Null (Zero) Matrix	All elements are zero.
Row Matrix	Only one row exists.
Column Matrix	Only one column exists.
Square Matrix	Number of rows = Number of columns
Diagonal Matrix	All non-diagonal elements are zero.
Identity (Unit) Matrix	All non-diagonal elements are zero, and all diagonal elements are one.
Scalar Matrix	All non-diagonal elements are zero, and all diagonal elements are equal.
Triangular Matrix	Elements on either side of the diagonal are zero.
Symmetric Matrix	Square matrix (A) = Transpose of the Matrix (AT)
Skew-symmetric Matrix	Square Matrix (A) = Negative of transpose of the Matrix (- AT)
Singular Matrix	Modulus of Matrix is zero (|A| = 0).
Non-singular Matrix	Modulus of Matrix is not zero (|A| ≠ 0).
Hermitian Matrix	Each mirror element across the diagonal is a complex conjugate.
Orthogonal Matrix	The product of a matrix and its transpose is the identity matrix.
Idempotent Matrix	The square of the matrix is the same matrix.

A Matrix in which all elements are zero is called a null matrix or zero matrix. It is denoted by O. the mathematical representation for Zero matrix is $A={\left[a_{ij}\right]}_{m\times n}$  such that $a_{ij}=0$  for all i & j.

For example,

• [0   0   0] is a zero matrix of order 1×3
• $\left[\begin{array}{cc}\hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill \end{array}\right]$ is a zero matrix of order 2×2
• $\left[\begin{array}{ccc}\hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \end{array}\right]$ is a zero matrix of order 3×2

 

Identification of the Zero Matrix

Zero matrices are very easy to identify; if all the elements are 0 in any matrix, no matter the order, it will be considered a Zero Matrix.

 

Applications of the Zero Matrix

• You can use a zero matrix as a null element in matrix addition: A + O = A – O = A
• A zero matrix can represent the absence of a relationship or no connection in a particular context.

Row Matrix: A matrix with only one row of elements is known as row matrix. Mathematically, it is represented as a matrix of order (1×n): A = ${\left[a_{ij}\right]}_{1\times n}$ .

For example,

• A = [-1    √5     7]  is a row matrix of order 1×3.
• $B=\left[\begin{array}{cc}a& b\end{array}\right]$ is a row matrix of order 1×2.

Column Matrix: A matrix which has only one column is called a column matrix. In general, a column matrix has the order of m × 1. The mathematical form of a column matrix is A = ${\left[a_{ij}\right]}_{m\times 1}$

For example,

• A = $\left[\begin{array}{cc}\hfill 1\hfill & \hfill 5\hfill \end{array}\right]$ is a column matrix of order 2x1.
• $B=\left[\begin{array}{c}a\\ b\\ c\end{array}\right]$ is a column matrix of order 3x1.

Identification of Row & Column Matrices

Any matrix that has only one row of elements is a row matrix (resembling a horizontal line). A matrix with only one column is a column matrix (resembling a vertical line).

 

Application of Row and Column Matrices

• Often these matrices are used to represent data (3D coordinate points or vectors) in a single row or column.
• These type of matrices can be used to display system of equations and solve using matrix concepts.

If the number of rows in a matrix are equal to the number of columns, the matrix is called a square matrix. Let’s assume both number of columns and rows are n, then the order of square matrix is n x n.

Mathematical form of a square matrix of order n: A = ${\left[a_{ij}\right]}_{n\times n}$

For example,

• A = [ 1    2  is a square matrix of order 2 x 2.

                 -1   4] 

• $B=\left[\begin{array}{ccc}a& b& c\\ j& k& l\\ p& q& r\end{array}\right]$ is a square matrix of order 3 x 3.

Identification of a Square Matrix

You just need to count the number of rows and columns; if they are equal, it is a square matrix.

 

Application of a Square Matrix

• You can use square matrices to solve linear equations by finding the inverse of the square matrix formed to get unknown values.
• Cryptography: Square matrices are used in encryption and decryption algorithms to secure data.
• Researchers also use square matrices to represent relationships in networks and encryption algorithms.

A rectangular matrix is a matrix where the number of rows is not equal to the number of columns. It is essentially a matrix that does not form a square. 

For example, A = $\left[\begin{array}{ccc}\hfill 12\hfill & \hfill -7\hfill & \hfill 9\hfill \\ \hfill 0\hfill & \hfill 13\hfill & \hfill -4\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 14\hfill \end{array}\right]$ is a rectangular matrix of order 3 $\times$ 2

Identification of a Rectangle Matrix

Count the number of rows and columns; if they are not equal, it is a rectangle matrix.

 

Application of a Rectangle Matrix

Rectangle matrices are usually used in data representation where various elements are involved. Rectangle matrix are also used in computer graphics.

A square matrix in which all the non-diagonal elements are zero, and the diagonal elements are non-zero constants. Mathematically, the diagonal matrices are represented in a standard form:

A = ${\left[a_{ij}\right]}_{n\times n}$ if $a_{ij}=0$ , for $i\ne j$ .

The elements $a_{11},a_{22},\dots ,a_{nn}$ are called diagonal elements. The line along which diagonal elements lie is called the principal diagonal.

A diagonal matrix of order n, having diagonal elements $d_1,d_2,\dots ,d_n$ is denoted by diag $\left[d_1,d_2,\dots ,d_n\right]$ .

For example, A = $\left[\begin{array}{ccc}\hfill 2\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 3\hfill \end{array}\right]$ is a diagonal matrix of order 3.

Identification of Diagonal Matrix

• Firstly, check if it is a square matrix. Since a diagonal matrix is always a square matrix.
• Then check for two things: first, any of the diagonal elements must be non-zero. All non-diagonal elements must be zero.

 

Application of Diagonal Matrix

• You can use diagonal matrices to simplify matrix operations. Diagonal matrices make multiplication and exponentiation simpler.
• In modern computing, diagonal matrices are widely used to represent in computer graphics. 

A square matrix  A = ${\left[a_{ij}\right]}_{n\times n}$ is called an identity matrix or unit matrix if all the non-diagonal elements are zero and diagonal elements are one.

In simple words, identity matrix in which all diagonal elements are one is called identity or unit matrix. The identity matrix of order n is denoted by $I_n$ .

$I_n$ = $a_{ij}=\{\begin{array}{l}0ifi\ne j\\ 1ifi=j\end{array}$ .

For Example;

• $I_2=\left[\begin{array}{cc}\hfill 1\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 1\hfill \end{array}\right]$ is identity matrix of order 2
• $I_3=\left[\begin{array}{ccc}\hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill \end{array}\right]$ are identity matrices of order 3

 

Identification of Identity Matrix

Check if all the non-diagonal elements are 0, and diagonal elements are one only such as $\left[\begin{array}{ccc}\hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill \end{array}\right]$ .

 

Application of Identity Matrix

You can use identity matrix as we use 1 in algebraic operation. That multiplication with identity matrix will not change the matrix (AI= A).