# 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 $$B \ A = I$$.