A rectangular arrangement of numbers into rows and columns is known as a **matrix**. The horizontal lines are called rows and the vertical lines are called columns. The matrix is enclosed by [] or ().

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Let A be any matrix. It can be written as A = [a_{ij}]_{m × n}

\[A=[a_{ij}]_{m\times n}=\begin{bmatrix} a_{11} & a_{12} & … & … & a_{1n}\\ a_{21} & a_{22} & … & … & a_{2n}\\ … & … & … & … & …\\ … & … & … & … & …\\ a_{m1} & a_{m2} & … & … & a_{mn} \end{bmatrix}\]

where, a_{ij} represents the element of row ‘i’ and column ‘j’. m × n represents the order of the matrix. It states that there are m rows and n columns in matrix A. Here i = 1, 2, 3, ……, m and j = 1, 2, 3, ……, n.

The plural of matrix is matrices. When we talk about more than one matrix then we called matrices.

## Types of Matrix

The special types of matrix are

- Row matrix
- Column matrix
- Null matrix
- Square matrix
- Identity matrix or Unit matrix
- Diagonal matrix
- Scalar matrix
- Singular matrix
- Non-singular matrix
- Upper triangular matrix
- Lower triangular matrix
- Idempotent matrix
- Involutory matrix
- Nilpotent matrix
- Real matrix
- Complex matrix

### 1. Row matrix

A matrix having only one row is called row matrix. For example,

\[\begin{bmatrix} 9 & 0 & -0.35 & 12 & 8 \end{bmatrix}\]

### 2. Column matrix

A matrix having only one column is called column matrix. For example,

\[\begin{bmatrix} 8\\ 1\\ 2\\ 0 \end{bmatrix}\]

### 3. Null matrix

A matrix that has all of its elements equal to zero is called null matrix. It is also known as zero matrix.

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

You can read more about null matrix here.

### 4. Square matrix

A matrix that has equal number of rows and columns is called square matrix. For example,

\[\begin{bmatrix} 3 \end{bmatrix}, \begin{bmatrix} 2 & 3\\ 4 & 5 \end{bmatrix}, \begin{bmatrix} 4 & 6 & 7\\ 0 & 4 & 8\\ -4 & -9 & 0 \end{bmatrix}\]

You can read more about square matrix here.

### 5. Identity matrix

A square matrix that has all of its diagonal elements equal to one and all of its off-diagonal elements equal to zero, is called identity matrix. It is also known as **unit matrix**. For example,

\[\begin{bmatrix} 1 & 0\\ 0 & 1 \end{bmatrix}, \begin{bmatrix} 1 & 0 & 0\\ 0 & 1 & 0\\ 0 & 0 & 1 \end{bmatrix}\]

You can read more about identity matrix or unit matrix here.

### 6. Diagonal matrix

A square matrix that has all of its off-diagonal elements equal to zero, is called diagonal matrix. For example,

\[\begin{bmatrix} 1 & 0\\ 0 &9 \end{bmatrix}, \begin{bmatrix} 1 & 0 & 0\\ 0 & 6 & 0\\ 0 & 0 & -2 \end{bmatrix}\]

You can read more about diagonal matrix here.

### 7. Scalar matrix

A square matrix that has all of its diagonal elements equal and all of its off-diagonal elements equal to zero, is called scalar matrix. It is a special case of diagonal matrix in which all diagonal elements are equal. For example,

\[\begin{bmatrix} 2 & 0\\ 0 & 2 \end{bmatrix}, \begin{bmatrix} -5 & 0 & 0\\ 0 & -5 & 0\\ 0 & 0 & -5 \end{bmatrix}\]

You can read more about scalar matrix here.

### 8. Singular matrix

A square matrix whose determinant is equal to zero, is called singular matrix. For example,

\[\begin{bmatrix} 1 &2 \\ 3& 6 \end{bmatrix}, \begin{bmatrix} 3 &8 &1 \\ -4& 1 &1 \\ -4&1 & 1 \end{bmatrix}\]

You can read more about singular matrix here.

### 9. Non-Singular matrix

A square matrix whose determinant is not equal to zero, is called non-singular matrix. For example,

\[\begin{bmatrix} 1 &0 \\ 0& 1 \end{bmatrix}, \begin{bmatrix} 1 &2 &3 \\ 4& 5 &6 \\ 7&8 & 9 \end{bmatrix}\]

You can read more about non-singular matrix here.

### 10. Upper triangular matrix

A square matrix whose lower off-diagonal elements are equal to zero, is called upper triangular matrix. For example,

\[\begin{bmatrix} 1 &2 \\ 0& -4 \end{bmatrix}, \begin{bmatrix} 1 &2 &3 \\ 0& 5 &6 \\ 0&0 & 9 \end{bmatrix}\]

You can read more about upper triangular matrix here.

### 11. Lower triangular matrix

A square matrix whose upper off-diagonal elements are equal to zero, is called lower triangular matrix. For example,

\[\begin{bmatrix} 1 &0 \\ 3& 6 \end{bmatrix}, \begin{bmatrix} 1 &0 &0 \\ 8& 5 &0 \\ 2&-1 & 9 \end{bmatrix}\]

You can read more about lower triangular matrix here.

### 12. Idempotent matrix

A square matrix **P** is said to be idempotent matrix if P^{2} = P. For example,

\[\begin{bmatrix} 0 &0 \\ 0& 0 \end{bmatrix}, \begin{bmatrix} 1 &0 \\ 0& 1 \end{bmatrix}, \begin{bmatrix} 2 &-2 &-4 \\ -1& 3 &4 \\ 1&-2 & -3 \end{bmatrix}\]

You can read more about idempotent matrix here.

### 13. Involutory matrix

A square matrix **P** is said to be involutory matrix if P^{2} = I, where I is the identity matrix. For example,

\[\begin{bmatrix} 4 &-1 \\ 15& -4 \end{bmatrix}, \begin{bmatrix} 1 &0 \\ 0& 1 \end{bmatrix}, \begin{bmatrix} 4 &3 &3 \\ -1& 0 &-1 \\ -4&-4 & -3 \end{bmatrix}\]

You can read more about involutory matrix here.

### 14. Nilpotent matrix

A square matrix **P** is said to be nilpotent matrix of class z if P^{z} = O, where O is the null matrix. For example,

\[B=\begin{bmatrix} 0 & 1 \\ 0& 0 \end{bmatrix}\]

It is a nilpotent matrix of class 2 i.e. B^{2} = O

\[D=\begin{bmatrix} 1 &1 &3 \\ 5& 2 &6 \\ -2&-1 & -3 \end{bmatrix}\]

It is a nilpotent matrix of class 3 i.e. D^{3} = O

You can read more about nilpotent matrix here.

### 15. Real matrix

A matrix that has all elements taking real values is called a real matrix.

Based on the relationship between matrix and its transpose, a real matrix can be classified into three types i.e.

- Symmetric matrix
- Skew symmetric matrix
- Orthogonal matrix

#### a. Symmetric matrix

A square matrix P is said to be symmetric matrix if P^{T} = P.

You can read more about symmetric matrix here.

#### b. Skew symmetric matrix

A square matrix P is said to be skew-symmetric matrix if P^{T} = -P.

You can read more about skew symmetric matrix here.

#### c. Orthogonal matrix

A square matrix P is said to be orthogonal matrix if P^{T} = P^{-1}.

You can read more about orthogonal matrix here.

### 16. Complex matrix

A matrix that has atleast one element taking complex value is called a complex matrix.

Based on the relationship between matrix and its conjugate transpose, a complex matrix can be classified into three types i.e.

- Hermitian matrix
- Skew hermitian matrix
- Unitary matrix

#### a. Hermitian Matrix

A complex square matrix P is said to be hermitian matrix if P^{θ} = P.

You can read more about hermitian matrix here.

#### b. Skew hermitian matrix

A complex square matrix P is said to be skew-hermitian matrix if P^{θ} = -P.

You can read more about skew hermitian matrix here.

#### c. Unitary matrix

A complex square matrix P is said to be unitary matrix if P^{θ} = P^{-1}.

You can read more about unitary matrix here.