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.

Solution

According to the question,

$\begin{bmatrix}3&5\end{bmatrix}$

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