A rectangular array of m x n numbers in the form of rows and columns is called a matrix. The numbers are enclosed by [] or () symbols. Here we will learn the definitions, examples, properties, and types of matrics. Also, check the solved example questions and more useful details about the matrix in the following sections.

## Definition of a Matrix

A matrix is defined as the rectangular arrangement or array of numbers or functions in the form of horizontal lines (rows) and vertical lines (columns) and is subject to certain rules of operation. Matrices are denoted by the capital letters of the alphabet.

**Example:**

The example of a matrix is given here.

\( A =\left[

\begin{matrix}

3 & 4 & 9\cr

12 & 11 & 35 \cr

\end{matrix}

\right]

\)

In the above matrix A, the numbers 3, 4, 9, 12, 11, and 35 are the elements of the matrix. The number 3 belongs to the 1st row and 1st column so it is called (1, 1)th element of the matrix A. As matrix A has 2 rows and 3 columns, the order of the matrix is 2 x 3. other important topics on matrices are provided below.

- Position of an Element in a Matrix
- Classification of Matrices
- Problems on Classification of Matrices
- Square Matrix
- Row Matrix
- Column Matrix
- Null Matrix
- Equal Matrices
- Identity (or Unit) Matrix
- Triangular Matrix
- Addition of Matrices
- Addition of Two Matrices
- Properties of Addition of Matrices
- Negative of a Matrix
- Subtraction of Matrices
- Subtraction of Two Matrices
- Scalar Multiplication of a Matrix
- Multiplication of a Matrix by a Number
- Properties of Scalar Multiplication of a Matrix
- Multiplication of Matrices
- Multiplication of Two Matrices
- Problems on Understanding Matrices
- Worksheet on Understanding Matrix
- Worksheet on Addition of Matrices
- Worksheet on Matrix Multiplication
- Worksheet on Matrix

### Types of Matrices

The different types of matrices and their definitions are given here.

**Symmetric Matrix:**A square matrix A = [a_{ij}] is called a symmetric matrix if a_{ij}= a_{ji}, for all i, j.**Skew-Symmetric Matrix**: A matrix A = [a_{ij}] is called skew-symmetric matrix is a_{ij}= -a_{ji}.**Orthogonal matrix**: If AA^{T}= In = A^{T}A**Hermitian and skew-Hermitian matrix**: A = A^{θ}(A^{θ}represent conjugate transpose). A^{θ}= -A is called skew-Hermitian matrix.**Idempotent Matrix**: If A^{2}= A**Involuntary materix:**If A^{2}= I or A^{-1}= A**Nilpotent Matrix:**A square matrix A is nilpotebt, if A^{p}= 0, p is an interger.

### Operations on Matrices

Matrix operations involve three algebra operations which are the addition of matrices, subtraction of matrices and multiplication of matrices. The details of each are given here.

**Matrix Addition:**

If A[a_{ij}]_{mxn} and B[b_{ij}]_{mxn} are two matrices having the same order, then their sim is A + B also a matrix. Every element of matrix A are added to the corresponding element of matrix B and the sum matrix will also have the same order.

The sum of 2 x 2 matrices is given as:

\( \left[

\begin{matrix}

a1 & a2 \cr

a3 & a4 \cr

\end{matrix}

\right]

\) + \( \left[

\begin{matrix}

b1 & b2 \cr

b3 & b4 \cr

\end{matrix}

\right]

\) = \( \left[

\begin{matrix}

a1 + b1 & a2 + b2 \cr

a3 + b3 & a4 + b4\cr

\end{matrix}

\right]

\)

**Subtraction of Matrices:**

If A and B are any two matrices having the same order, then A – B = A + (-B). The difference of two 2 x 2 matrices is given as:

\( \left[

\begin{matrix}

a1 & a2 \cr

a3 & a4 \cr

\end{matrix}

\right]

\) – \( \left[

\begin{matrix}

b1 & b2 \cr

b3 & b4 \cr

\end{matrix}

\right]

\) = \( \left[

\begin{matrix}

a1 – b1 & a2 – b2 \cr

a3 – b3 & a4 – b4\cr

\end{matrix}

\right]

\)

The matrix subtraction involves subtracting each element of one matrix from the corresponding element of the second matrix. The obtained matrix also have the same order.

**Matrix Multiplication:**

If A and B be any two matrices, then their product AB will be defined only when the number of columns in A is equal to the number of rows in B. If A = [a_{ij}]_{mxn} and B = [b_{ij}]_{nxp} then their product AB = C = [c_{ij}]_{mxp} and (AB)_{ij} = C_{ij} = ∑_{r=1}^{n} a_{ir}b_{rj}.

### Properties of Matrix

Matrix properties for addition operation are provided here. You can check the properties of matrices page to know other properties.

**1. Commutative Law:**

If A and B are two matrices, then A + B = B + A. For three matrices A, B, C, commutative law is if A + B = A + C then B + A = C + A and B = C.

**2. Associative Law:**

For any three matrices A, B, C the asssociative law states that (A + B) + C = A + (B + C).

**3. Identity Law:**

A matrix is added to the given matrix, and the sum is the same as the given matrix, called the identity matrix for addition.

A + O = O + A = A

Here O is the zero matrix.

**4. Additive Inverse:**

A matrix added to the given matrix to get the sum zero is called additive inverse.

A + (-A) = 0 = (-A) + A

### Examples on Matrix

**Example 1:
**If \( B =\left[

\begin{matrix}

8 & 5 & 4\cr

6 & 8 & 2 \cr

\end{matrix}

\right]

\). What is the order of matrix B?

**Answer:
**The order of matrix B is 2 × 3 because there are 2 rows and 3 columns in the matrix.

**Example 2:
**If a matrix has 8 elements, find the possible orders of the matrix.

**Answer:
**8 = 1 x 8

8 = 8 x 1

8 = 4 x 2

8 = 2 x 4

Therefore, the possible orders of the matrix are 1 x 8, 8 x 1, 4 x 2, 2 x 4.

### Frequently Asked Question’s on Matrix

**1. What is the best definition of a matrix?**

A matrix is a collection of numbers arranged into a fixed number of rows and columns. The numbers are real numbers. The numbers are enclosed by [] or () parenthesis.

**2. What is the matrix in real life?**

In real life, matrices are applied in the study of electrical circuits, quantum mechanics, optics. It is also helpful for the calculation of battery power outputs.

**3. What is another word for the matrix?**

Another word that defines a matrix is Array, grid, table or spreadsheet.