## Matrix

Subject: Optional Maths

#### Overview

A number arranged in rows and columns which enclosed in large brackets or round brackets with a rectangular arrangement is called the matrix. Matrix is a rectangular arrangement of numbers which contain elements in the form of row and column
##### Matrix

Definition

A number arranged in rows and columns which enclosed in large brackets or round brackets with a rectangular arrangement is called matrix. It is represented by capital letters . Here, its plural form is matrices .

For eg ;

A = $\begin{bmatrix} 2 & 3 \\ 1 & 4 \end{bmatrix}$ B = $\begin{bmatrix} 2 & 1 & 3 \\ 6 & 4 & 2 \end{bmatrix}$

C = $\begin {bmatrix} 4 & 2 \\ -3 & 2 \\ 1 & 0\end{bmatrix}$

Rows and colums of a matrix

Matrix is a rectangular arrangement of numbers which contain elements in the form of row and column .Here, a column is usually a vertical line and row is a horizontal line .

eg;

A = $\begin{bmatrix} 3 & 2 \\ 1 & 5\end{bmatrix}$

This matrix A has two rows and two columns .

Order of matrix

The number of rows and columns present in a matrix gives the order of a matrix .

eg;

A = $\begin {bmatrix} 3 & 1 & 2 \\ 5 & 4 & 6 \end {bmatrix}$

Matrix A has 2 rows and 3 columns So, its order is 2 x 3.And it i read as 2 by 3.

Types of Matrices.

1.Row Matrix

A matrix having only one row is known as a matrix row . It can have any number of columns .

eg; A = $\begin {bmatrix} -3 & 1 & 2\end{bmatrix}$

It has only one row .

B = $\begin{bmatrix} -1 & 2\end {bmatrix}$

It has only one row .

Column Matrix

2.A matrix having only one column but any number of rows is known as a column matrix .

eg; A =$\begin{bmatrix} -1 \\2 \\ 3 \end{bmatrix}$ B = $\begin {bmatrix} 1 \\ 2 \end{bmatrix}$

Here, A and B both are matrics have only one column .

3.Rectangular Matrix

A matrix having unequal number of rows and columns is known as a rectangular matrix .But,the number of rows and columns should not be equal .

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

Here matrix A has 2 rows and 3 columns is unequal .So, it is rectangular matrix .

B = $\begin{bmatrix} 1 & -4 \\ 2 & 5 \\ 3 & 6 \end{bmatrix}$

4.Zero Matrix

A matrix having each of the elements zero ( 0 ) is known as a null or zero matrix .

O = $\begin {bmatrix} 0 & 0 & 0 \\ 0 & 0 & 0\end{bmatrix}$

It is a null matrix which is of order 2 x 3.

Square Matrix

A matrix whose number of rows and columns are equal is called a square matrix .

A = $\begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}$ B = $\begin{bmatrix} 3 & 2 & 1 \\ 4 & 6 & 5 \\ 7 & 9 & 8 \end{bmatrix}$

Then, In matrix A and B the number of rows and columns both are equal . So, they are squar matrix.

5.Equal Matrices

A pair of matrices having same order and equal corresponding elements are said to be equal matrices .

eg

A = $\begin{bmatrix} 2 & 3 \\ 4 & 5 \end{bmatrix}$ B = $\begin{bmatrix} 2 & 3 \\ 4 & 5\end{bmatrix}$

Here, matrices A and B are said to be equal matrices as their order is same as well as their corresponding elements are equal .

6.Operation on matrices

In this level we shall only see how matrices are added and subtracted . Let's look at the following examples.

A = $\begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}$ and B = $\begin{bmatrix} 5 & 6 \\ 7 & 8\end{bmatrix}$

A + B = ?

Here, as the order of both the matrices is 2 x 2. we can add them

then, A + B = $\begin {bmatrix} 1 + 5 & 2 + 6 \\ 3 + 7 & 4 +8 \end{bmatrix}$ [ We add the corresponding elements ]

=$\begin{bmatrix} 6 & 8\\ 10 & 12 \end{bmatrix}$

Again,

A = $\begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}$ C = $\begin{bmatrix} 5 \\ 6 \end{bmatrix}$

A + C = ? [ Order is not equal so, no addition is possible ]

= $\begin{bmatrix} 1 & 2 \\ 3 & 4\end{bmatrix}$ + $\begin{bmatrix} 5 \\ 6 \end{bmatrix}$

As the order of these two matrices A and C are not the same we can not add them .

Now,

B - A = ?

so, B - A

= $\begin{bmatrix} 5 - 1 & 6 - 2 \\ 7 - 3 & 8 - 4 \end{bmatrix}$

= $\begin{bmatrix} 4 & 4 \\ 4 & 4\end{bmatrix}$

##### Things to remember
• The column is usually a vertical line and row is a horizontal line .
• Matrix is represented by capital letters .
• Its plural form is matrices .
• It includes every relationship which established among the people.
• There can be more than one community in a society. Community smaller than society.
• It is a network of social relationships which cannot see or touched.
• common interests and common objectives are not necessary for society.
##### Solving Variables in Equal Matrices (Equivalent Matrices)

Solution

A order 2$\times$3

It means matrix   A should have  2 rows and 3 columns.

A = $\begin {bmatrix} 1&2&5\\ 2&4&6\\\end {bmatrix}$

Solution

Here, C order 3$\times$3

It means  matrix C must have 3 rows and 3 columns.

C=  $\begin {bmatrix} 2 & 5& 8\\ 1 & 4 & 7 \\ 3 & 6 & 9\\\end {bmatrix}$

Solution

A= $\begin {bmatrix}2&5 \\ -5&3 \\ 9&7 \\\ \end {bmatrix}$

This is rectangular matrix.

It's order is 3 $\times$ 2.

Solution

P= $\begin{bmatrix}1&2&3\\4&5&6\\7&8&9\end{bmatrix}$

Here, Rows present in matrix P= 3

Columns present in matrix  p = 3

Therefore, Order of Matrix = 3 $\times$ 3

Solution

$\begin {bmatrix} a&5\\6&b\end {bmatrix}$ =  $\begin {bmatrix} 7&c\\d&4\end {bmatrix}$

Here, As the matrices are equal, their corresponding elements are also equal.

So, a = 7, b = 4, c = 5 and d = 6.

Solution

Since the matrices are equal, their corresponding elements are also equal. So,

x - 5 = 0

or, x = 5

Again,
y + 3 =  3

y = 0

Hence, x = 5 and y= 0

Solution

Since the matrices are equal their corresponding elements are also equal.

So,  2 = 4c
or, c = $\frac{1}{2}$

Again,
3a = 6
or, a = $\frac{6}{3}$
or, a =  2

Similarly,
2b = $\frac{1}{3}$
b = $\frac{1}{3×2}$
b= $\frac{1}{6}$

And,
3d = $\frac{1}{8}$

d= $\frac{1}{3×8}$

d=$\frac{1}{24}$

Therefore, a= 2, b= $\frac{1}{6}$, c= $\frac{1}{2}$ and d= $\frac{1}{24}$.

Solution

Since the matrices are equal, their corresponding elements are also equal.

So, 5x + 1 = 11

or, 5x = 10

∴ x= 2

Again,
3y - 2 = -5

or, 3y = -5 + 2

or, 3y = -3

∴ y = -1

Hence, x =  2 and y = -1.

Solution

Since the matrices are equal, their corresponding elements are also equal.

So, x-2 = 3x-5

or, x-3x = -5 + 2

or, -2x = -3

x = $\frac{3}{2}$

and

or,  y+3 = 9

or, y = 9 - 3

or,  y = 6

Hence, x = $\frac{3}{2}$ and y= 6.

Solution

If we arrange these numbers in a proper arrangement, we can arrange them as

$\begin{bmatrix}40&50&60\\70&80&90\end{bmatrix}$ or $\begin{bmatrix}40&50\\60&70\\80&90\end{bmatrix}$

Where in the first case rows determine class and columns determine terminal exam.

Similarly, in the second case the  rows determine the terminal exam and columns determine the class.

A Matrix is a rectangular arrangement of numbers arranged in rows and columns enclosed within large brackets.