‘Matrix’ is a singular while ‘matrices’ is a plural form.

**Matrices**

**1. Matrices**

‘Matrix’
is a singular while ‘matrices’ is a plural form. Matrix is a rectangular array
of numbers systematically arranged in rows and columns within brackets. In a
matrix, if the number of rows and columns are equal, it is called a square
matrix.

**2. Determinants**

For every
square matrix, there exists a determinant. This determinant is an arrangement
of same elements of the corresponding matrix into rows and columns by enclosing
vertical lines.

is a determinant of the matrix A
denoted by *A* .

The value
of the determinant is expressed as a single number.

Calculation
of the value of determinant for a 2 x 2 matrix is shown below

Calculation
of determinant value for a 3 x 3 matrix is shown below

The value
of determinant is 40.

**3. Cramer’s
Rule**

Cramer’s
rule provides the solution of a system of linear equations with ‘**n**’ variables and ‘**n**’ equations. It helpsto arrive at a unique solution of a system of
linear equations with as many equations as unknowns.

a_{11}x
+ a_{12} y + a_{13} z = b_{1}

a_{21}x
+ a_{22} y + a_{23} z = b_{2}

a_{31}x
+ a_{32} y + a_{33} z = b_{3}

then

__Example: 12.8__

Find the
value of x and y in the equations by using Cramer’s rule.

x + 3y = 1 and

3x - 2y = 14

Then the equations in the matrix form :

**Key
Note**

If the determinant ∆=0 , then solution does not exist.

**Answer checking:**

Substituting
in equation the values of x and y,

4+3(-1) = 1,

3(4) –
2(-1)= 14

__Example: 12.9__

Find the
solution of the system of equations.

5x_{1}
+ 3x_{2} = 30

6x_{1}
- 2x_{2} = 8

__Solution:__

The
coefficient and the constant terms are given below for the equations

__Example: 12.10__

Find the
solution of the equation system

7x_{1}
- x_{2} - x_{3} = 0

10x_{1}
- 2x_{2} + x_{3} = 8

6x_{1}
+ 3x_{2} - 2x_{3} = 7

__Solution:__

**4. Application
in Economics**

__Example 12.11__

Mr.Anbu,
purchased 2 pens, 3 pencils and 1 note book. Mr.Barakath , purchased 4 pens, 3
pencils and 2 notebooks. Mr.Charles purchased 2 pens, 5 pencils and 3
notebooks. They spent ** Rs.**32, ** Rs.**52 and ** Rs.**60
respectively. Find the price of a pen, a pencil and a note book.

__Solution:__

Let x be
the price of a pen, y be the price of a pencil and z be the price of a
notebook, In

equations:

2x + 3y + 1z = 32,

4x + 3y + 2z = 52,

2x + 5y +3z = 60

In matrix
form

**Answer
checking**

2(5)+3(4)+1(10)=32

4(5)+3(4)+2(10)=52

2(5)+5(4)+3(10)=60

**Think
and Do**

Fathima, purchased 6 pens and 5 Pencils spending Rs.49, Rani purchased 3 Pens and 4 pencils spending Rs.32. What is the price of a pen and pencil?

Solution : Price of a pen = Rs.4

Price of a pencil = Rs.5

Study Material, Lecturing Notes, Assignment, Reference, Wiki description explanation, brief detail

11th Economics : Chapter 12 : Mathematical Methods for Economics : Matrices |

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