All the important types of matrices are discussed below.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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 $$ A^T = \begin{bmatrix} 1 & 3 \\ 2 & 4 \\ \end{bmatrix} $$
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). \((A + B)^T = A^T + B^T\)
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: $$ A = \begin{bmatrix} 1 & 2 \\ 2 & 1 \\ \end{bmatrix} $$ $$ A^T = \begin{bmatrix} 1 & 2 \\ 2 & 1 \\ \end{bmatrix} $$
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: $$ A = \begin{bmatrix} 0 & 2 \\ -2 & 0 \\ \end{bmatrix} $$ $$ A^T = \begin{bmatrix} 0 & -2 \\ 2 & 0 \\ \end{bmatrix} $$
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 \(A^2 = I\)
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 \(A \ A^T = I\).
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 \(A^2 = A\)
A matrix is said to be a Lower Triangular matrix if all the elements above the leading diagonal are zero.
Lower Triangular Matrix Example: $$ A = \begin{bmatrix} 1 & 0 & 0 \\ 2 & 3 & 0 \\ 4 & 5 & 6 \\ \end{bmatrix} $$
A matrix is said to be an Upper Triangular matrix if all the elements below the leading diagonal are zero.
Upper Triangular Matrix Example: $$ A = \begin{bmatrix} 1 & 2 & 3 \\ 0 & 4 & 5 \\ 0 & 0 & 6 \\ \end{bmatrix} $$
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.
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.
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 \(B \ A = I\).