# Types of Matrices

A matrix is a rectangular array of numbers, arranged in rows and columns. Here are two of such matrices:

$\begin{pmatrix}1&2&3\\4&5&6\end{pmatrix},\quad \begin{pmatrix}1&2\\3&4\\5&6\end{pmatrix}.$

In this wiki, we are only going to discuss different types of matrices.

#### Contents

## Some Basic Types of Matrices

Here we will discuss square matrix, horizontal matrix, vertical matrix, row matrix, column matrix, null matrix, diagonal matrix, and scalar matrix.

**Square Matrix:**

Take a rectangular matrix $A = (a_{ij})_{m \times n}$ of order $m \times n$. If $m = n,$ then matrix $A$ is said to be a square matrix. In other words, if in a matrix the number of rows and columns are equal, then the matrix is said to be a square matrix. Given below are a few examples of square matriices:

$\begin{pmatrix} a & b \\ c & d \\ \end{pmatrix},\quad \begin{pmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \\ \end{pmatrix},\quad \begin{pmatrix} x & y \end{pmatrix}.$

**Horizontal Matrix:**

Take a rectangular matrix $A = (a_{ij})_{m \times n}$ of order $m \times n$. If $m < n,$ then matrix $A$ is said to be a horizontal matrix. In other words, if in a matrix the number of rows is less in number than the number of columns, then that matrix is said to be a horizontal matrix. Given below are a few examples of horizontal matriices:

$\begin{pmatrix} a & b \\ x & y \\ c & d \\ \end{pmatrix},\quad \begin{pmatrix} 2 \\ 4 \\ 12 \\ \end{pmatrix},\quad \begin{pmatrix} 1 & 0 \\ 3 & 9 \\ 7 & 2 \\ 5 & 8 \\ \end{pmatrix}.$

**Vertical Matrix:**

Take a rectangular matrix $A = (a_{ij})_{m \times n}$ of order $m \times n$. If $m > n,$ then matrix $A$ is said to be a vertical matrix. In other words, if in a matrix the number of rows is more in number than the number of columns, then that matrix is said to be a vertical matrix. Given below are a few examples of horizontal matriices:

$\begin{pmatrix} a & b & c \\ x & y & d \\ \end{pmatrix},\quad \begin{pmatrix} 2 & 4 & 12 \end{pmatrix},\quad \begin{pmatrix} 1 & 0 & 7 & 5 \\ 3 & 9 & 2 & 8 \\ 7 & 2 & 3 & 10 \\ \end{pmatrix}.$

**Row Matrix:**

Take a rectangular matrix $A = (a_{ij})_{m \times n}$ of order $m \times n$. If $m = 1,$ then matrix $A$ is said to be a row matrix. In other words, if in a matrix there is only one row present, then that matrix is said to be a row matrix. Given below are a few examples of row matriices:

$\begin{pmatrix} 1 \\ 7 \\ 4 \\ \end{pmatrix},\quad \begin{pmatrix} p \\ q \\ r \\ \end{pmatrix},\quad \begin{pmatrix} 4 \\ 3 \\ \end{pmatrix}.$

**Column Matrix:**

Take a rectangular matrix $A = (a_{ij})_{m \times n}$ of order $m \times n$. If $n = 1,$ then matrix $A$ is said to be a column matrix. In other words, if in a matrix there is only one coloum present, then that matrix is aid to be a column matrix. Given below are a few examples of row matrices:

$\begin{pmatrix} 1 & 7 & 4 \\ \end{pmatrix},\quad \begin{pmatrix} p & q & r \\ \end{pmatrix},\quad \begin{pmatrix} 4 & 3 \\ \end{pmatrix}.$

**Null Matrix:**

In any matrix if each and every element is $0$, it is called a null matrix or **zero matrix**. Given below are 2 null matrices of order $2 \times 2$ and $3 \times 3,$ respectively:

$\begin{pmatrix} 0 & 0 \\ 0 & 0 \\ \end{pmatrix},\quad \begin{pmatrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ \end{pmatrix}.$

**Diagonal Matrix:**

A square matrix in which all non-diagonal elements are $0$ is called a diagonal matrix. The diagonal elements also may be $0$ or may not be $0,$ but at least there should be one non-zero element. Given below are 2 such matrices:

$\begin{pmatrix} 1 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 3 \\ \end{pmatrix},\quad \begin{pmatrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 1 \\ \end{pmatrix}.$

**Scalar Matrix:**

A diagonal matrix in which all principle diagonal elements are equal is called a scalar matrix. Given below are a few examples of scalar matrices:

$\begin{pmatrix} 3 & 0 & 0 \\ 0 & 3 & 0 \\ 0 & 0 & 3 \\ \end{pmatrix},\quad \begin{pmatrix} a & 0 & 0 \\ 0 & a & 0 \\ 0 & 0 & a \\ \end{pmatrix}.$

## Identity Matrix

A diagonal matrix in which all principal diagonal values are equal to 1 is called as an **identity matrix** or a **unit matrix**. It is denoted by the letter $I$. Given below is an Identity matrix of order $3 \times 3$:

$I_3 = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\ \end{pmatrix},$

where the subscript $3$ in $I_3$ tells us that it is an identity matrix of order $3 \times 3$. Similarly,

$I_2 = \begin{pmatrix} 1 & 0 \\ 0 & 1 \\ \end{pmatrix}.$

**Properties:**

You can compare an identity matrix $I$ in matrices and normal $1$ in mathematics. They both play a similar role as follows:

- An identity matrix raised to a power of any positive integer will not change but remain as it is: $\begin{aligned} I^2 &= I \\ I^3 &= I \\ I^4 &= I \\ &\vdots \\ I^n &= I. \end{aligned}$
- When any matrix is multiplied with an identity matrix, the values in the matrix will not change: $I \times A = A.$

## Special Types of Matrices

Here we will discuss matrices which are idempotent, nilpotent, involuntary, periodic, and singular.

**Idempotent Matrix:**

Consider a matrix $A = (a_{ij})_{m \times n}$ of order $m \times n$. If $A^2 = A$, then matrix $A$ is said to be Idempotent.

The best example for an idempotent matrix is the unit matrix itself:

$I = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\ \end{pmatrix},\quad I^2 = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\ \end{pmatrix}.$

Verify whether $A = \begin{pmatrix} 2 & -2 & -4 \\ 1 & 3 & 4 \\ 1 & -2 & -3 \\ \end{pmatrix}$ is idempotent or not.

To check whether $A$ is idempotent or not, we should first find $A^2:$$\begin{aligned} A^2 &= \begin{pmatrix} 2 & -2 & -4 \\ 1 & 3 & 4 \\ 1 & -2 & -3 \\ \end{pmatrix} \begin{pmatrix} 2 & -2 & -4 \\ 1 & 3 & 4 \\ 1 & -2 & -3 \\ \end{pmatrix} \\\\ &= \begin{pmatrix} 4 - 2 + 4 & -4 - 6 - 8 & 8 - 8 + 12 \\ 2 + 3 + 4 & - 2 + 9 - 8 & 4 + 12 + 12 \\ 2 - 2 - 3 & -2 - 6 + 6 & 4 - 8 - 9 \\ \end{pmatrix} \\\\ &= \begin{pmatrix} 6 & -18 & 12 \\ 9 & -1 & 28 \\ -3 & -2 & -3 \\ \end{pmatrix}. \end{aligned}$

Clearly, $A^2 \neq A$. Therefore, $A$ is not idempotent. $_\square$

Suppose that matrices $A$ and $B$ are both idempotent. Also, $A$ and $B$ are both commutative. Verify whether matrix $AB$ is idempotent or not.

Given below are the conditions mentioned in the question:$\begin{aligned} A^2 &= I \\ B^2 &= I \\ AB &= BA. \end{aligned}$

To check whether $AB$ is idempotent or not, we should first find $(AB)^2:$

$\begin{aligned} (AB)^2 &= (A{\color{blue}B})({\color{blue}A}B) \\ &= A({\color{blue}BA})B \\ &= A({\color{blue}AB})B \qquad \qquad (\text{as AB = BA}) \\ &= (A{\color{blue}A})({\color{blue}B}B)\\ &= \underbrace{A^2}_{\text{equal to A}} \cdot \underbrace{B^2}_{\text{equalt to B}} \\ &= AB. \end{aligned}$

Clearly, $(AB)^2 = AB.$ Therefore, $AB$ is an idempotent matrix. $_\square$

If $A$ is an idempotent matrix, then verify whether matrix $I - A$ is Idempotent or not, where $I$ is a unit matrix of the same order as $A$.

We are given that $A^2 = A$. As usual, to check whether $(I - A)$ is idempotent or not, we should first find $(I - A)^2:$$\begin{aligned} (I - A)^2 &= (I - A)(I - A) \\ &= I^2 - IA - AI + A^2 \\ &= I - A - \cancel{A} + \cancel{A} \qquad \qquad \big(\text{as } A^2 = A , I^2 = I\big) \\ &= I - A. \end{aligned}$

Clearly, $(I - A)^2 = I - A$. Therefore, $(I - A)$ is idempotent matrix. $_\square$

## Triangular Matrix

## Transpose Matrix

## Symmetric and Skew-symmetric Matrix

## Orthogonal Matrix

**Cite as:**Types of Matrices.

*Brilliant.org*. Retrieved from https://brilliant.org/wiki/types-of-matrices/