Types of Matrices
A matrix is a rectangular arrangement of elements in rows and columns. Different types of matrices are classified based on their size, structure, and properties. Below are some of the most common types of matrices with examples.
1. Row Matrix
A row matrix is a matrix that has only one row and multiple columns.
Example:
\[ R = \begin{bmatrix} 4 & 7 & 2 \end{bmatrix} \]
Here, \( R \) is a \( 1 \times 3 \) matrix.
2. Column Matrix
A column matrix is a matrix that has only one column and multiple rows.
Example:
\[ C = \begin{bmatrix} 5 \\ 8 \\ 3 \end{bmatrix} \]
Here, \( C \) is a \( 3 \times 1 \) matrix.
3. Square Matrix
A square matrix has the same number of rows and columns.
Example:
\[ S = \begin{bmatrix} 2 & 4 & 6 \\ 3 & 5 & 7 \\ 1 & 8 & 9 \end{bmatrix} \]
Here, \( S \) is a \( 3 \times 3 \) square matrix.
4. Diagonal Matrix
A diagonal matrix is a square matrix where all the non-diagonal elements are zero.
Example:
\[ D = \begin{bmatrix} 5 & 0 & 0 \\ 0 & 3 & 0 \\ 0 & 0 & 7 \end{bmatrix} \]
Here, only the diagonal elements are non-zero.
5. Scalar Matrix
A scalar matrix is a diagonal matrix where all the diagonal elements are equal.
Example:
\[ S = \begin{bmatrix} 4 & 0 & 0 \\ 0 & 4 & 0 \\ 0 & 0 & 4 \end{bmatrix} \]
All diagonal elements are the same (4).
6. Identity Matrix
An identity matrix is a square matrix where all the diagonal elements are 1 and all non-diagonal elements are 0.
Example:
\[ I_3 = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix} \]
The identity matrix is often denoted as \( I_n \), where \( n \) is the size.
7. Zero Matrix
A zero matrix (or null matrix) is a matrix in which all elements are zero.
Example:
\[ Z = \begin{bmatrix} 0 & 0 & 0 \\ 0 & 0 & 0 \end{bmatrix} \]
Here, \( Z \) is a \( 2 \times 3 \) zero matrix.
8. Upper Triangular Matrix
An upper triangular matrix is a square matrix where all elements below the main diagonal are zero.
Example:
\[ U = \begin{bmatrix} 3 & 5 & 2 \\ 0 & 7 & 6 \\ 0 & 0 & 8 \end{bmatrix} \]
All elements below the diagonal are zero.
9. Lower Triangular Matrix
A lower triangular matrix is a square matrix where all elements above the main diagonal are zero.
Example:
\[ L = \begin{bmatrix} 4 & 0 & 0 \\ 5 & 1 & 0 \\ 2 & 3 & 9 \end{bmatrix} \]
All elements above the diagonal are zero.
10. Symmetric Matrix
A symmetric matrix is a square matrix that is equal to its transpose (\( A = A^T \)).
Example:
\[ S = \begin{bmatrix} 1 & 2 & 3 \\ 2 & 4 & 5 \\ 3 & 5 & 6 \end{bmatrix} \]
The elements are symmetric across the main diagonal.
11. Skew-Symmetric Matrix
A skew-symmetric matrix is a square matrix where the transpose is equal to the negative of the original matrix (\( A^T = -A \)).
Example:
\[ A = \begin{bmatrix} 0 & -2 & -3 \\ 2 & 0 & -5 \\ 3 & 5 & 0 \end{bmatrix} \]
Each element satisfies \( a_{ij} = -a_{ji} \).
12. Singular and Non-Singular Matrices
- A matrix is singular if its determinant is zero.
- A matrix is non-singular if its determinant is non-zero.
Example of a singular matrix:
\[ S = \begin{bmatrix} 2 & 4 \\ 1 & 2 \end{bmatrix} \]
\(\text{det}(S) = (2 \times 2) – (4 \times 1) = 4 – 4 = 0\), so the matrix is singular.