Types of Matrices:


Types of Matrices:


All the important types of matrices are discussed below.


Horizontal Matrix:


A matrix is said to be a horizontal matrix if it has less-number of rows than the number of columns.

Horizontal Matrix Example: $$ A = \begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ \end{bmatrix} $$ It is a horizontal matrix of order \(2 \times 3\) because it has 2 rows and 3 columns.


Vertical Matrix:


A matrix is said to be a vertical matrix if it has more rows than the columns.


Vertical Matrix Example: $$ A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \\ 5 & 6 \\ \end{bmatrix} $$ It is a vertical matrix of order \(3 \times 2\) because it has 3 rows and 2 columns.


Row Matrix or Row Vector:


A matrix with only one row is called a row matrix or row vector.


Row Matrix Example: $$ A =\begin{bmatrix} 1 & 2 & 3 & 4 \\ \end{bmatrix} $$ It is a row matrix of order \(1 \times 4\) because it has only 1 row and 4 columns.


Column Matrix or Column Vector:


A matrix with only one column is called a column matrix or column vector.


Column Matrix Example: $$ A = \begin{bmatrix} 1 \\ 2 \\ 3 \\ \end{bmatrix} $$ It is a column matrix of order \(3 \times 1\) because it has 3 rows and only 1 column.


Square Matrix:


A matrix is said to be a square matrix if it has an equal number of rows and columns.


Square Matrix Example: $$ A = \begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \\ \end{bmatrix} $$ It is a square matrix of order \(3 \times 3\) because it has an equal number of rows and columns.


Zero Matrix or Null Matrix:


A matrix is called a zero matrix or a null matrix if all the elements are zero. The zero matrix or null matrix is denoted by O.


Zero Matrix Example: \(O_1 = \begin{bmatrix} 0 & 0 & 0 \\ \end{bmatrix}\), \(O_2 = \begin{bmatrix} 0 \\ 0 \\ 0 \\ \end{bmatrix}\), \(O_3 = \begin{bmatrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ \end{bmatrix}\)

These all the matrices are the zero matrices or null matrices of orders \(1 \times 3\), \(3 \times 1\), and \(3 \times 3\) respectively.


Diagonal Matrix:


A matrix is called a diagonal matrix if all its non-diagonal elements are zero.


Diagonal Matrix Example: $$ A = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 3 \\ \end{bmatrix} $$ It is a diagonal matrix of order \(3 \times 3\) because all the non-diagonal elements of this matrix are zero.


Note: If \(A_{n \times n}\) is a square matrix of order n, then leading diagonal elements will be \(a_{11}, a_{22}, a_{33}.......a_{nn}\). For example if \(A = \begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \\ \end{bmatrix}\) then leading diagonal elements will be 1, 5, 9.


Unit or Identity Matrix:


A matrix is called a Unit matrix or identity matrix if all the elements of the leading diagonal are 1 and all the non-diagonal elements are zero. The identity matrix is denoted by 'I'.


Identity Matrix Example: $$ I = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\ \end{bmatrix} $$ It is an identity matrix of order \(3 \times 3\) because all the leading diagonal elements are 1 and all the non-diagonal elements are zero.


Scalar Matrix:


A matrix is called a scalar matrix if all the elements of the leading diagonal are equal and all the non-diagonal elements are zero.


Scalar Matrix Example: $$ A = \begin{bmatrix} 2 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 2 \\ \end{bmatrix} $$ It is a scalar matrix of order \(3 \times 3\) because all the leading diagonal elements are equal and all the non-diagonal elements are zero.


Equality of two matrices:


Two matrices will be equal if

(i). The order of both the matrices are the same and,

(ii). The corresponding elements of both matrices are the same.


Example: $$ A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \\ \end{bmatrix} $$ $$ B = \begin{bmatrix} 1 & 2 \\ 3 & 4 \\ \end{bmatrix} $$ These matrices A and B are equal because the order of both the matrices A and B are the same (\(2 \times 2\) order matrices), and the corresponding elements of both the matrices also the same.


Transpose Matrix:


A matrix \(A = a_{ij}\) is said to be a transpose matrix if the rows and columns of this matrix will be interchanged with each other. The transpose matrix of A is denoted by \(A^{'}\) or \(A^T\).


Transpose Matrix Example: If $$ A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \\ \end{bmatrix} $$ Then the transpose matrix will be $$ A^T = \begin{bmatrix} 1 & 3 \\ 2 & 4 \\ \end{bmatrix} $$


Properties of Transpose Matrix:

For any matrix these properties are true.


(1). \((A^T)^T = A\)

(2). \((AB)^T = A^T \ B^T\)

(3). \((kA)^T = k \ A^T\) {k is a constant}

(4). \((A + B)^T = A^T + B^T\)


Symmetric Matrix:


A matrix \(A = a_{ij}\) is said to be symmetric matrix if \(A^T = A\) or \(a_{ij} = a_{ji}\). In other words, if after interchanging the rows and columns of any matrix, we will still get the same matrix then it is called a Symmetric matrix.


Sysmmetric Matrix Example: $$ A = \begin{bmatrix} 1 & 2 \\ 2 & 1 \\ \end{bmatrix} $$ $$ A^T = \begin{bmatrix} 1 & 2 \\ 2 & 1 \\ \end{bmatrix} $$


Skew Sysmmetric Matrix:


A matrix \(A = a_{ij}\) is said to be a skew symmetric matrix if \(A^T = -A\) or \(a_{ij} = -a_{ji}\) and the leading diagonal elements are zero.


Skew Sysmmetric Matrix Example: $$ A = \begin{bmatrix} 0 & 2 \\ -2 & 0 \\ \end{bmatrix} $$ $$ A^T = \begin{bmatrix} 0 & -2 \\ 2 & 0 \\ \end{bmatrix} $$


Involutory Matrix:


A matrix is said to be an involutory matrix if \(A^2 = I\).


Involutory Matrix Example: If $$ A = \begin{bmatrix} 1 & 0 \\ 0 & 1 \\ \end{bmatrix} $$ then $$ A^2 = \begin{bmatrix} 1 & 0 \\ 0 & 1 \\ \end{bmatrix} \ \begin{bmatrix} 1 & 0 \\ 0 & 1 \\ \end{bmatrix} $$ $$ A^2 = \begin{bmatrix} 1 & 0 \\ 0 & 1 \\ \end{bmatrix} $$ Hence \(A^2 = I\)


Orthogonal Matrix:


A matrix is said to be an Orthogonal matrix if \(A \ A^T = I\).


Orthogonal Matrix Example: If $$ A = \begin{bmatrix} 1 & 0 \\ 0 & -1 \\ \end{bmatrix} $$ then $$ A^T = \begin{bmatrix} 1 & 0 \\ 0 & -1 \\ \end{bmatrix} $$ Hence $$ A \ A^T = \begin{bmatrix} 1 & 0 \\ 0 & -1 \\ \end{bmatrix} \ \begin{bmatrix} 1 & 0 \\ 0 & -1 \\ \end{bmatrix} $$ $$ A \ A^T = \begin{bmatrix} 1 & 0 \\ 0 & 1 \\ \end{bmatrix} $$ We get \(A \ A^T = I\).


Idempotent Matrix:


A matrix is said to be an Idempotent matrix if \(A^2 = A\).


Idempotent Matrix Example: If $$ A = \begin{bmatrix} 1 & 0 \\ 0 & 1 \\ \end{bmatrix} $$ then $$ A^2 = \begin{bmatrix} 1 & 0 \\ 0 & 1 \\ \end{bmatrix} \ \begin{bmatrix} 1 & 0 \\ 0 & 1 \\ \end{bmatrix} $$ $$ A^2 = \begin{bmatrix} 1 & 0 \\ 0 & 1 \\ \end{bmatrix} $$ Hence \(A^2 = A\)


Lower Triangular Matrix:


A matrix is said to be a Lower Triangular matrix if all the elements above the leading diagonal are zero.


Lower Triangular Matrix Example: $$ A = \begin{bmatrix} 1 & 0 & 0 \\ 2 & 3 & 0 \\ 4 & 5 & 6 \\ \end{bmatrix} $$


Upper Triangular Matrix:


A matrix is said to be an Upper Triangular matrix if all the elements below the leading diagonal are zero.

Upper Triangular Matrix Example: $$ A = \begin{bmatrix} 1 & 2 & 3 \\ 0 & 4 & 5 \\ 0 & 0 & 6 \\ \end{bmatrix} $$

Singular Matrix:


A matrix is said to be a Singular matrix if the value of the determinant of any square matrix is zero.


Singular Matrix Example: If $$ A = \begin{bmatrix} 2 & 3 \\ 1 & \frac{3}{2} \\ \end{bmatrix} $$ $$ |A| = \begin{vmatrix} 2 & 3 \\ 1 & \frac{3}{2} \\ \end{vmatrix} $$ $$ |A| = 2 \times \frac{3}{2} - 3 \times 1 $$ $$ |A| = 0 $$ As the value of \(|A|\) is zero, hence it is a singular matrix.


Non-Singular Matrix:


A matrix is said to be a Non-Singular matrix if the value of the determinant of any square matrix is not zero.


Non-Singular Matrix Example: If $$ A = \begin{bmatrix} 2 & 3 \\ 1 & 5 \\ \end{bmatrix} $$ $$ |A| = \begin{vmatrix} 2 & 3 \\ 1 & 5 \\ \end{vmatrix} $$ $$ |A| = 2 \times 5 - 3 \times 1 $$ $$ |A| = 7 $$ As the value of \(|A|\) is not zero, hence it is a non-singular matrix.


Inverse Matrix:


A matrix is said to be an inverse matrix if there are two square matrices A and B of the same order multiplied with each other, and the answer will be an identity matrix \(AB = BA = I\). Here B is called an inverse matrix of A, and A is called an inverse matrix of B. The inverse matrix is denoted like this \(A^{-1}\).


Inverse Matrix Example: If $$ A = \begin{bmatrix} 2 & 3 \\ 1 & 2 \\ \end{bmatrix} $$ then $$ B = \begin{bmatrix} 2 & -3 \\ -1 & 2 \\ \end{bmatrix} $$ Hence $$ A \ B = \begin{bmatrix} 2 & 3 \\ 1 & 2 \\ \end{bmatrix} \ \begin{bmatrix} 2 & -3 \\ -1 & 2 \\ \end{bmatrix} $$ $$ A \ B = \begin{bmatrix} 1 & 0 \\ 0 & 1 \\ \end{bmatrix} $$ We get \(A \ B = I\). $$ B \ A = \begin{bmatrix} 2 & -3 \\ -1 & 2 \\ \end{bmatrix} \ \begin{bmatrix} 2 & 3 \\ 1 & 2 \\ \end{bmatrix} $$ $$ A \ B = \begin{bmatrix} 1 & 0 \\ 0 & 1 \\ \end{bmatrix} $$ We get \(B \ A = I\).