Types of Matrices with Examples

In this article, we will define different types of matrices along with examples.

Matrix: A matrix is a form of a rectangular array of m rows and n columns. These elements are multiplied during matrix multiplication keeping in mind some basic matrix multiplication properties.

Elements of a matrix: All the numbers inside a matrix are called elements of the matrix.

Types of Matrices are:
  1. Row Matrix.
  2. Column matrix.
  3. Null matrix.
  4. Square Matrix.
  5. Diagonal matrix.
  6. Scalar matrix.
  7. Identity or Unit matrix.
  8. Idempotent matrix.
  9. Periodic matrix.
  10. Triangular matrix.
  11. Involuntary matrix.
  12. Nilpotent matrix.
  13. Sub-Matrix.
  14. Principal sub-matrix.
  15. Singular matrix.
  16.  Non-singular matrix.

1. Row Matrix. A matrix which has only one row is said to be a row matrix. For example;

\[\begin{bmatrix}
1&2&3\\
\end{bmatrix}\]

2. Column matrix. Column matrix is a matrix which has only one column. For example;

\[\begin{bmatrix}
1\\
2\\
9\\
\end{bmatrix}\]

Transpose of a row matrix is a column matrix while as the transpose of a column matrix is a row matrix.

3. Null matrix. Null matrix is that matrix whose all the elements are zero. For example;

\[\begin{bmatrix}
0&0&0\\
0&0&0\\
0&0&0\\
\end{bmatrix}\]

4. Square Matrix. A square matrix is a matrix in which the number of rows is equal to the number of columns. For Example;

\[\begin{bmatrix}
1&2&4\\
5&1&9\\
2&6&1\\
\end{bmatrix}\]

5. Diagonal matrix. A diagonal matrix is a square matrix in which all the elements except the elements in leading diagonal are zero. For Example;

\[\begin{bmatrix}
7&0&0\\
0&5&0\\
0&0&1\\
\end{bmatrix}\]

6. Scalar matrix. A scalar matrix is a diagonal matrix in which all the diagonal elements are equal. For example;

\[\begin{bmatrix}
7&0&0\\
0&7&0\\
0&0&7\\
\end{bmatrix}\]

7. Identity or Unit matrix. An identity matrix is a diagonal matrix in which all the diagonal elements are unity. For example;
\[\begin{bmatrix}
1&0&0\\
0&1&0\\
0&0&1\\
\end{bmatrix}\]

8. Idempotent matrix. A matrix [A] is said to be idempotent matrix if its square “[A]2” is equal to the matrix [A] itself. If [A] is a matrix of order (m×n) and [A]2 = [A], then the matrix [A] is called as an idempotent matrix. For example;

\[If\:A=\begin{bmatrix}
1&0\\
0&1\\
\end{bmatrix}\]

\[and\:[A]^2=\begin{bmatrix}
1&0\\
0&1\\
\end{bmatrix}\]

The matrix [A] is said to be an idempotent matrix.

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9. Periodic matrix. A matrix [A] is said to be periodic if [A]x+1 = [A], where “x” is any positive integer.

10. Triangular matrix. A triangular matrix is of two types:

  • Upper triangular matrix: A square matrix is said to be upper triangular matrix if it all the elements above the leading diagonal are zero. For example;

\[\begin{bmatrix}
1&0&0\\
2&7&0\\
6&8&9\\
\end{bmatrix}\]

  • Lower triangular matrix: A square matrix is said to be a lower triangular matrix if its all the elements below the leading diagonal are zero. For example;

\[\begin{bmatrix}
1&7&5\\
0&7&8\\
0&0&9\\
\end{bmatrix}\]

11. Involuntary matrix. A matrix is said to be involuntary if its square is equal to unit or identity matrix.

12. Nilpotent matrix. A matrix is said to be nilpotent if [A]x = 0 or null matrix, where x is any positive integer.

13. Sub-Matrix. when we omit some rows and columns from a given (m×n) matrix, then the remaining matrix is called a sub-matrix. For example;

\[If\:A=\begin{bmatrix}
1&5&2\\
6&7&8\\
2&9&4\\
\end{bmatrix}\]

\[and\:B=\begin{bmatrix}
1&5\\
6&7\\
\end{bmatrix}\]

The matrix B is said to be the sub-matrix of matrix A.

14. Principal sub-matrix.  Principal sub-matrix is a square submatrix of a square matrix [A] whose diagonal elements are also the diagonal elements of the matrix [A].

15. Singular matrix. It is a matrix whose determinant is zero.

16. Non-Singular matrix. It is a matrix whose determinant is zero. If it is a square matrix then it is invertible.

These are all the different types of matrices. If you find any error in our article please comment below. You can also contact us in the contact section.

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