Types of Matrices

A matrix is actually an arrangement of numbers, symbols, or expressions in rows and columns. Every element in a matrix is put at a specific position. Matrices are used in mathematics, physics, engineering, and computer science. Now, let's take a closer look at different types of matrices in simple words.

1.0The Most Common Types of Matrices

Although there are different types of matrices are of many types, here are some of the most common matrices that are used in not only maths but also in physics. 

1. Row Matrix

A row matrix is a matrix that has only one row. The number of columns can be any positive integer, but the number of rows is always one.

Properties:

• It has the form 1×n, where n is the number of columns.
• A row matrix could be added or subtracted from other matrices of the same order.

Example: Here, matrix A has 1 Row and 3 Columns

2. Column Matrix

A column matrix is the opposite of a row matrix. It has only one column. The number of its rows can be any positive integer, but the columns are always one in number.

Properties:

• It has the form n  1, where n is the number of rows.
• It is commonly used to represent vectors in mathematics.

Example: Matrix B has 3 Rows and 1 Column.      

3. Square Matrix

It is a type of matrix that has the same number of rows and columns. A square matrix can have orders like 2 2, 33, etc. 

Properties:

• The number of rows equals the number of columns.
• Square matrices have unique properties like having a determinant and an inverse (if the determinant is non-zero).

Example: Matrix C is a 22  matrix.

4. Zero Matrix (Null Matrix)

The zero matrix has all the elements that are zero. It can be of any size.

Properties:

• It can be added to any matrix without changing the other matrix.
• The product of any matrix and a zero matrix is always a zero matrix.

Example: Matrix Z is a 33 zero matrix.

2.0Special Types of Matrices

Here are some special types of matrices in maths that have some specific properties. 

1. Identity Matrix

An identity matrix is a special type of square matrix in which all the diagonal elements are equal to 1 (One), and all other elements are 0 (Zero). The identity matrix is often denoted by I_n, where n is the order of the matrix.

Properties:

• An identity matrix is always square.
• It acts as a multiplicative identity, meaning that any matrix multiplied by the identity matrix gives the same matrix.

Example: Matrix I₃ is a 33 identity matrix.

2. Diagonal Matrix

A diagonal matrix is a square matrix in which all elements in the matrix outside the major diagonal are zero. The elements of the main diagonal may have any values. 

Properties:

• Only the diagonal elements are non-zero.
• If all diagonal elements are equal, it is a scalar matrix.

Example: Matrix D is a 33 diagonal matrix.

3. Scalar Matrix

A scalar matrix is a special case of a diagonal matrix where all the diagonal elements are the same.

Properties:

• A scalar matrix is always a square matrix.
• Scalar matrices are used in scaling transformations in geometry.

Example: Matrix S is a scalar matrix because all diagonal elements are 6

4. Symmetric Matrix

A symmetric matrix is such type of square matrix that is equal to its transpose. That is, if A is a symmetric matrix, then. A=AT.

Properties:

• It should be square.
• Symmetric matrices often appear in optimisation problems and physics.

Example: Matrix A is symmetric because A=AT.

5. Skew-Symmetric Matrix

A skew-symmetric matrix is a square matrix that is the negative of its transpose. That is, if A is skew-symmetric, then A=-AT.

Properties:

• All diagonal elements of a skew-symmetric matrix are zero.
• Skew-symmetric matrices are used to describe rotational transformations.

Example: Matrix N is skew-symmetric because N=-NT.

6. Idempotent Matrix

A matrix A is said to be idempotent if A² =A. That is, the matrix multiplied by itself results in the same matrix. Idempotent matrices have applications in many fields, like projection operators in linear algebra.

7. Nilpotent Matrix

A matrix A is said to be nilpotent if A^k=0 for some positive integer k, where 0 is the zero matrix. Nilpotent matrices have all eigenvalues equal to zero, and their powers eventually become the zero matrix.

8. Upper and Lower Triangular Matrix

• Upper Triangular Matrix: A square matrix in which all the entries below the main diagonal are zero. Example:

• Lower Triangular Matrix: A square matrix in which all the entries above the main diagonal are zero. Example:

9. Singular and Non-singular Matrix

Singular Matrix: A square matrix that is non-invertible. Its determinant is zero (det(A)=0). 

Non-singular Matrix: A square matrix that is invertible, and the determinant of the matrix is non-zero.   (det(A)≠0).

10. Hermitian Matrix

A matrix A is Hermitian if A=A^H, where A^H is the conjugate transpose of A. All the eigenvalues of a Hermitian matrix are always real, and its diagonal elements are real. Hermitian matrices arise in quantum mechanics and many areas of physics.

3.0Solved Examples

Problem 1: Show that a matrix which is both symmetric and skew-symmetric is a zero matrix.

Solution: A matrix Let A be symmetric if the matrix is equal to its transpose.  

A = A^T …….(1)

This means that all the elements of i^th row are equal to j^th row and j^th column is equal to i^th column of A. 

a_ij = a_ji

For the matrix let A to be a skew-symmetric matrix, A needs to be equal to the negative of the transpose of the matrix. 

A = – A^T …….(2)

This also implies that a_ij = a_ji

Now, equating both equations 

A = A^T = – A^T

A +  A^T = 0

A + A = 0 

2A = 0 

A = 0

Therefore, A must be the zero matrix. This proves that a matrix which is both symmetric and skew-symmetric is necessarily a zero matrix.

Problem 2: Let 

 Then show that A² – 4A + 7I = O. 

Using this result,  calculate A⁵ also. 

Solution: 

Problem 3: Find values of a and b if A = B, where

Solution: Given that A = B. So, every element of A will be equal to B

a + 4 = 2a + 2 

2a – a = 4 – 2 

a = 2. 

b² + 2 = 3b 

b² – 3b + 2 = 0 

b² – 2b – b + 2 = 0

b(b–2) – 1(b–2) = 0

(b–1)(b–2) = 0 

b = 1, 2.