What are the Different Types of Matrices?

 There are different types of Matrices. Here they are -

1) Row matrix

2) Column matrix

3) Null matrix

4) Square matrix

5) Diagonal matrix

6) Upper triangular matrix

7) Lower triangular matrix

8) Symmetric matrix

9) Skew -symmetric matrix

10) Horizontal matrix

11) Vertical matrix

12) Identity matrix

Let's discuss the different types of matrices in mathematics, types of matrix in detail, matrices definition and types.

1. Null Matrix

If in a matrix all the elements are zero then it is called a zero matrix and it is generally denoted by 0. Thus, A=[aij]m×n is a zero-matrix if aij=0 for all iand j.

The first matrix O is a 2×2 matrix with all the elements equal to zero and the second matrix O is a 3×3 matrix with all the elements equal to zero.

O=[0000],O=⎡⎣⎢000000000⎤⎦⎥

2. Triangular Matrix

A square matrix is known to be triangular if all of its elements above the principal diagonal are zero then the triangular matrix is known as a lower triangular matrix or all of its elements below the principal diagonal are zero then the triangular matrix is known as upper triangular matrix.

⎡⎣⎢100260359⎤⎦⎥

The matrix given above is a 3×3 upper triangular matrix.

The matrix given below is an example of a 3×3 lower triangular matrix.

⎡⎣⎢123045006⎤⎦⎥

3. Vertical Matrix

A matrix of order m×n is known as a vertical matrix of m>n, where m is equal to the number of rows and n is equal to the number of columns.

Matrix Example

⎡⎣⎢⎢⎢21325164⎤⎦⎥⎥⎥

In the matrix example given below the number of rows (m)=4, whereas the number of columns (n)=2.Therefore, this makes the matrix a vertical matrix.

4. Horizontal Matrix

A matrix of order m×nis known as a horizontal matrix if n>m, where m is equal to the number of rows and n is equal to the number of columns.

Matrix Example

[12253141]

In the matrix example given below the number of rows (m) = 2, whereas the number of columns (n) = 4. Therefore, we can say that the matrix is a horizontal matrix.

5. Row Matrix

A matrix that has only one row is known as a row matrix. Thus A = aijm×n is a row matrix if m is equal to 1.

It is known so because it has only one row and the order of a row matrix will hence always be equal to 1×n.

Example of a Row matrix,

A=[469],B=[721925]

In the matrix example given above, matrix A has only one row and so matrix B has one row, therefore both matrices A and B are row matrices.

6. Column Matrix

A matrix that has one column is known as a Column matrix. Thus A = aijm×n is a column matrix if n is equal to 1.

It is known so because it has only one column and the order of a column matrix will hence always be equal to m×1

Example of a Column matrix,

A=⎡⎣⎢348⎤⎦⎥,B=⎡⎣⎢⎢⎢4982⎤⎦⎥⎥⎥

In the matrix example given above, matrix A has only one column and matrix B has one column, therefore both matrices A and B are column matrices.

7. Diagonal Matrix

If all the elements of the matrix, except the principal diagonal in any given square matrix, is equal to zero, it is known as a diagonal matrix. Thus a square matrix A = aij

is a diagonal matrix if aij=0, when i is not equal to j

For example,

⎡⎣⎢200030004⎤⎦⎥

The example given above is a diagonal matrix as it has elements only in its diagonal.

8. Symmetric Matrix

A square matrix A=[aij] is known as a Symmetric matrix if aij=aji, for all i,j values.

For example,

A=⎡⎣⎢123245352⎤⎦⎥

9. Skew -Symmetric Matrix

A square matrix A=[aij] is a skew-symmetric matrix if aij=aji, for all values of i,j. Thus, in a skew-symmetric matrix all diagonal elements are equal to zero.

For example,

⎡⎣⎢0−2−12031−30⎤⎦⎥

10. Identity Matrix

If all the elements of a principal diagonal in a diagonal matrix are 1 , then it is called a unit matrix. A unit matrix of order n$\mathrm{n}$ can be denoted by In. Thus, a square matrix A
[aij]m×n is an identity matrix if all its diagonals have value 1.

 For example, A=⎡⎣⎢100010001⎤⎦⎥