*Matrices concept knowledge is most important in mathematics. It made simplification of linear equations easy. Matrices have major applications in a wide range like it is used in spreadsheet programs, in business areas and science that is in budgeting, sales, cost estimation and also in sociology, psychology, etc.* Introducing of the matrices tool helped in solving algebraic problems very easily. Let’s first understand the concept of matrices and then their application in algebra.

## Matrices Definition in Matrices Concept Basics

Matrices are an ordered rectangular array consisting of numbers or functions or elements represented in rows and columns.

\begin{align*}

A=\begin{bmatrix} -3 & 4 \\ 5 & 6 \\ 7 & -8 \end{bmatrix} \ \\ \\ B=\begin{bmatrix} 1+y & y^{2} & 2 & \\ sin x & cosx & \tan x \end{bmatrix}

\end{align*}

### Row Elements or Rows

The elements represented in a horizontal line inside the matrix comes under rows.

### Column Elements or Columns

The elements represented in a vertical line inside the matrix comes under columns.

### Order of Matrices

Any matrix consisting of m rows and n columns is represented in the order m×n. This m×n is called order of the matrix and the matrix can be called as m×n matrix.

Order of a matricx can be represented as **number of rows × number of columns**

### Representation of m×n Matrices in Matrices Concept

The above matrix consists of ‘m’ rows and ‘n’ columns.

Simply matrix can be represented as A=[a_{ij}]_{m×n} where 1≤i≤m, 1≤j≤n i,j ∈ N

‘i’ indicates row elements number and elements in i^{th} row are represented as a_{i1}, a_{i2}, a_{i3}………..a_{in}.

‘j’ indicates row elements number and elements in j^{th} row are represented as a_{1j}, a_{2j}, a_{3j}………..a_{mj}.

a_{ij} is the element whose poaition in matrix is i^{th} row and j^{th} column.

### Examples for Representation of Matrices for different Orders

\begin{align*}

\begin{aligned}A=\begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix} _{2\times 2} \ \end{aligned} \end{align*}

\begin{align*}

\begin{aligned}B=\begin{bmatrix} 1 & 3 & 0 & 4 \\ 2 & -1 & 1 & 6 \end{bmatrix} _{2\times 4} \end{aligned} \end{align*}

\begin{align*}

\begin{aligned}C=\begin{bmatrix} 1 & 3 \\ 7 & 9 \\ 0 & 4 \end{bmatrix} _{3\times 2} \\ \\ \\ D=\begin{bmatrix} 1 \\ 2 \\ 3 \end{bmatrix} _{3\times 1} \ \ E=\begin{bmatrix} 1&2 & 3 \end{bmatrix} _{1\times 3} \end{aligned}

\end{align*}

## Types of Matrices

### Row Matrix

A matrix which consists of elements in only one row is a row matrix.**Example:**

A=[-5 3 2 ]

‘A’ is a row matrix of order 1×4.

### Column Matix

A matrix that consists of elements in only one column is a column matrix.**Example:**

\begin{align*}

B=\begin{bmatrix} -5 \\ \sqrt {2} \\ 5 \end{bmatrix}

\end{align*}

‘B’ is a column matrix of order 3 × 1

### Square Matrix

A matrix that has a same number of rows and columns is a square matrix.**Example:**

A matrix of order m × n is said to be a square matrix if m=n and is represented as order m × m or n × n

i,e A=[a_{ij}]_{m×m} or A=[a_{ij}]_{n×n}

\begin{align*}

A=\begin{bmatrix} 1 & 2 & 3 \\ 4 & -5 & \sqrt {6} \\ -7 & 8 & 9 \end{bmatrix}_{3\times 3}

\end{align*}

‘A’ is square matrix of order 3 × 3.

### Diagonal Matrix

If non-diagonal elements of a square matrix are zero then that matrix is a diagonal matrix.

A=[a_{ij}]_{m×m} is said to be diagonal matrix if a_{ij} =0, where i ≠ j**Example:**

\begin{align*}

A=\begin{bmatrix} -2 & 0 & 0 \\ 0 & 3 & 0 \\ 0 & 0 & \sqrt {2} \end{bmatrix}_{3\times 3}

\end{align*}

### Scalar Matrix

A diagonal matrix with all its diagonal elements equal is a scalar matrix.

A=[a_{ij}]_{m×m} is said to be a scalar matrix

if a_{ij} =0 where i ≠ j

a_{ij} =k where i=j, k is any value.**Example:**

\begin{align*}

A=\begin{bmatrix} \sqrt {2} & 0 & 0 \\ 0 & \sqrt {2} & 0 \\ 0 & 0 & \sqrt {2} \end{bmatrix}_{3\times 3} \ \\ \\ B=\begin{bmatrix} -5 & 0 \\ 0 & -5 \end{bmatrix}_{2\times 2}

\end{align*}

### Identity Matrix

A diagonal matrix with all its diagonal elements equal to 1 is an identity matrix.

Represented as I_{n} or simply I. Here ‘n’ is the order of identity matrix.

I_{n} = [a_{ij}]_{n×n} where a_{ij} =0 if i ≠ j a_{ij} =1 if i=j,

\begin{align*}

A=\begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}_{3\times 3} \ \ \ B=\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}_{2\times 2}

\end{align*} **NOTE:**

1. Every identiy matrix is a scalar matrix.

2. A singular matrix with k=1 is an identity matrix.

### Zero Matrix

A matrix with all its elements as zero is called a zero matrix or null matrix.

Represented as O= [O_{ij}]_{m×n} **Example:**

\begin{align*}

A=\begin{bmatrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{bmatrix}_{3\times 3} \ \ \ B=\begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix}_{2\times 2}

\end{align*}

### Transpose of a Matrix

A matrix formed by interchanging the rows and columns of a matrix [a_{ij}]_{m×n} is called transpose of matrix [a_{ij}]_{m×n} . It is represented by A^{T} or A^{|} .

Let A= [a_{ij}]_{m×n} then, A^{T} =[a_{ij}]_{n×m} **Example:**

\begin{align*}

A=\begin{bmatrix} 0 & 1 & 2 \\ 3 & 4 & 5 \end{bmatrix} \ then \ A^{T}=\begin{bmatrix} 0& 3 \\ 1 & 4 \\ 2 & 5 \end{bmatrix}

\end{align*}

### Symmetric Matrix

If transpose of a matrix [a_{ij}]_{m×n} is equal to the matrix itself then it is a symmetric matrix.

That is A^{T}=A ⇒ [a_{ij}] = [a_{ji}]

for all i & j \begin{align*}

A=\begin{bmatrix} 1 & \sqrt{2} & 3 \\ \sqrt{2} & -5 & 4 \\ 3 & 4 & 1 \end{bmatrix}_{3\times 3} \\ \\ is \ a \ symmetric \ matrix

\end{align*}

### Skew Symmetric Matrix

If transpose of a given matrix is equal to the additive identity of that matrix then the matrix is skew symmetric matrix.

That is A^{T}= -A ⇒ [a_{ij}] = – [a_{ji}]

if i=j then a_{ii} = a_{ii}

⇒ 2 a_{ii} =0

⇒ a_{ii} =0 For all ‘i’ values.

Therefore, Diagonal elements of a skew-symmetric matrix are zero’s.**Example:**

\begin{align*}

A=\begin{bmatrix} 0 & 1 & 2 \\ 1 & 0 & -4 \\ 2 & -4 & 0 \end{bmatrix}_{3\times 3} \\ \\ is \ a \ skew-symmetric \ matrix

\end{align*}

**NOTE:**

- ‘A’ is a square matrix

then**A+A**^{T}is a symmetric matrix and**A-A**^{T}is a skew-symmetric matrix. - Any square matrix can be represented as sum of a symmetric and a skew-symmetric matrix.
**A= ½(A+A**^{T}) + ½(A-A^{T})