Matrix is the most powerful tool in mathematics. We use matrices in certain branches of science as far as in economics, computer, psychology and management.

A matrix is an arrangement of numbers or symbols in rectangle array. The items in a matrix are called elements of matrix. Given below is an example of a matrix with eight elements:

$\begin{bmatrix} 1 &3 &4 & -1 \\ 2 &5 &-3 & 5 \end{bmatrix}$

With the help of matrices, we can solve a system of linear equation.

## What is a Matrix?

A matrix is a rectangular arrangement of numbers. We arrange numbers in rows(horizontal lines) and columns(vertical lines). In general, a matrix with m rows and n column is denoted by $A_{m\times n }$ and called $m\times n$ order matrix.

Order of a matrix (also called as the dimension of the matrix) represents the number of rows and column in a given matrix. So, $A_{m\times n }$ =$\left [c_{i,j}Â \right ]_{m\times n }$, where $c_{i,j}$ refers to the elements of matrix and subscript m and n refers to the row number and column number respectively.

The element $c_{i,j}$ belongs to the ith row and jthcolumn and is sometimes called the (i,j)th element of the matrix. Here i = 1, 2..... m and j = 1, 2, 3..... n.

In matrix, the number of columns and rows need not to be equal.

## Types of Matrices

In mathematics, there are different types of matrices which are classified according to their order as follows:

• Row Matrix
• Column Matrix
• Square Matrix
• Diagonal Matrix
• Symmetric and Skew-symmetric Matrix
• Null Matrix
• Unit Matrix

## Equal Matrices

Let A and B be two matrices of same order. These matrices are said to be equal if their corresponding elements are equal.

Let $A_{m\times n}$ = [ xij ] and $B_{m\times n}$ = [ yij ] are two matrices of the same order ($m\times n$) if $x_{ij} = y_{ij}$ for all i and j. So, we write A = B.

### Solved Examples

Question 1: A = $\begin{bmatrix} 1 &2 \\ 1 &2 \end{bmatrix}$ and B = $\begin{bmatrix} 1 &2 \\ 1 &2 \end{bmatrix}$
Find whether the matrices A ad B are equal.

Solution:
From the above example, it is clear that matrices A and B are of same order $(2\times 2)$ and their corresponding elements are equal. So, A and B are equal matrices.

Question 2: Find whether the following matrices are equal matrices. C = $\begin{bmatrix} 1 &3 &5 &7 \end{bmatrix}$ and D = $\begin{bmatrix} 8 &1 \\ 2 &3 \\ 4 &5 \end{bmatrix}$
Solution:
Here, order of C1x4 is not equal to the order of D2x3. So, C$\neq$D. By the use of the above property, we can find out the value of unknown variables.

Question 3: Find the value of x and y in the matrices A =$\begin{bmatrix} x &3 &6 \\ 1 &y-2 &9 \end{bmatrix}$ and B =$\begin{bmatrix} 2 &3 &6 \\ 1 &1 &9 \end{bmatrix}$
Solution:
It is clear from the above example that both matrices have the same order $(2\times3)$. So, we use the property which states that corresponding elements should be equal.
i.e. x = 2, 3 = 3, 6 = 6, 1 = 1, y - 2 = 1, 9 = 9.
So, we get x = 2 and y = 3.

## Matrix Operations

Just like operations on numbers, we use some operations on matrix also. They are addition, subtraction, multiplication etc.

Matrix Addition and Subtraction: Matrix addition and subtraction are possible only if the order of two matrices A & B are the same. We can add or subtract the corresponding elements of the matrices.
i.e. A $\pm$ B = [aij] $\pm$ [bij]

Matrix Multiplication: If $A_{p\times q}$ = [aij] and $B_{q\times r}$ = [bij] are two matrices, then AB is a matrix of order $p\times r$.
So, $(AB)_{p\times r}$ = [cij]

## Transpose of a Matrix

Let A = [aij] be a given matrix. If we interchange the rows and columns of matrix A, then the resultant matrix is known as transpose of matrix A and is denoted by AT, i.e. If A = [aij], then AT = [aij].

### Solved Example

Question: Find the transpose of the matrix $A_{3\times 3}$=$\begin{bmatrix} a &b &-c \\ -d &e &s \\ u &v &w \end{bmatrix}$
Solution:
$A^{T}$=$\begin{bmatrix} a &-d &u \\ b &e &v \\ -c &s &w \end{bmatrix}$

## Determinant of a Matrix

If A = [aij] is a square matrix, then the determinant whose elements are aij is called the determinant of A and is denoted by $\left | A \right |$.
Let A = $\begin{bmatrix} a_{11} &a_{12} \\ a_{21} &a_{22} \end{bmatrix}$
The determinant of A is as follows:
$\left | A \right |$ = (a11a22 - a21a12).

If A = [aij] is a square matrix, then to determine adjoint of matrix A, first we have to calculate the cofactor matrix of A and then the transpose matrix of that cofactor matrix. Thus, the resultant matrix is known as Adjoint matrix of the given matrix and is denoted by adj A.
A = [aij] The adjoint of matrix A is as follows:
adj A=$\left [c _{ij} \right ]^{T}$

## Inverse of a Matrix

If A be the given square matrix, then the inverse of A is written as A-1 and is defined as follows:
A -1 =$\frac{adjA}{\left | A \right |}$, here $\left | A \right | \neq 0$

## Matrix Row Operations

In mathematics, there are four basic operations on numbers namely addition, subtraction, multiplication and division. But in matrices,there are only three operation that we can apply on rows of a given matrix:
• Interchanging the two rows like $R_{1}\leftrightarrow R_{2}$
• Multiply any row by a scalar or number $3R_{1}$ or $4R_{2}$
• Multiply k times the elements of a row. Then, add and subtract p times the corresponding elements of another row, where k and p are real numbers.
These operations are called Elementary Row Transformation. We can easily understand these operations from the examples shown below:

Let A = $\begin{bmatrix} 2 &1 &3 &7 \\ 2 &6 &4 &8 \\ 1 &1 &0 &1 \end{bmatrix}$

Apply the following operation on matrix A, $R_{1}\leftrightarrow R_{3}$ (interchanging the rows)

B $\approx$ $\begin{bmatrix} 1 &1 &0 &1 \\ 2 &6 &4 &8 \\ 2 &1 &3 &7 \end{bmatrix}$

Apply the following operation on matrix B, $R_{2}\rightarrow$$\frac{1}{2}\times R_{2}$ (multiplying row by a number)

C $\approx$ $\begin{bmatrix} 1 &1 &0 &1 \\ 1 &3 &2 &4 \\ 2 &1 &3 &7 \end{bmatrix}$

Apply the following operation on matrix C, $R_{3}\rightarrow R_{3}-2R_{2}$ (operation between two rows)

D $\approx$ $\begin{bmatrix} 1 &1 &0 &1 \\ 1 &3 &2 &4 \\ 0 &-5 &-1 &-1 \end{bmatrix}$
Thus, after applying Elementary Transformation on A, we have obtained the matrix D.

## Properties of Matrices

Listed below are some of the properties of matrices.

Properties of addition: Let A and B are two matrices of the same order ($m\times n$) and c be any scalar.
 Commutative Property A + B = B + A Associative Property (A + B) + C = A + (B + C) Existence of an Additive Identity B + 0 = B = 0 + B Existence of an Additive Inverse B + (-B) = 0 Distributive Property c(A + B) = cA + cB

Properties of multiplication: Let $A_{l \times m}$, $B_{m \times n}$, $C_{n \times p}$ are three matrices and t,q are the scalars.

 Associative property A(BC) = (AB)C Distributive property A(B + C) = AB + BC Multiplicative Identity 1A = A Property for scalar multiplication (tq)A = t(qA) Existence of Multiplicative Inverse A (A-1) = In

Properties of Transpose:
• (AB)T = BTAT
• (A + B)T = AT + BT
• (AT)T = A
• (qA)T = qAT