Determinants

Determinants (सारणिक)

 Determinant of 2nd Order ||

|| is called determinant of 2nd

order. Here are called elements of the Similarly it can be expanded along and

determinant and its value is .  Rules to put ’ of – (Sign Scheme)
If be an element of a determinant then the sign of
Ex- Find the value of the determinants is determined by the expression ( ) .

|| | | Let | |

| | | Sign of () ()
Soln: | Sign of () ()
a. | () Sign of () ()
b. | Sign of () ()
Sign of () ()
c. | | Sign of () ()
Sign of () ()
Sign of () ()
Sign of () ()

() Sign Scheme of | | is
 Determinant of 3rd Order
| |
||

|| | | is called ||
determinant of 3rd order.

Note A determinant of 3rd order can be expended Q. Find the value of the determinants
through any row or any column thus it can be expended in
six different ways (3 rows + 3 columns) || ||

Let | | Soln:
a. |
Expanding through (row-1) || || |
| || ( )(
| ||
)(
)

|| b. | |
Expanding along i.e, 2nd row
| | || || |
| || ( )( ) ( )

||  Minor and Cofactor(उपसारणिक तथा सहखडं )
Expanding through i.e, 3rd row Minor:- If be an element of a determinant then its

| || | minor is nothing but the value of the determinant

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()

Determinants (सारणिक)

obtaining after deleting the row and the column in () ()
which lies from the original determinants. () ()
Ex- () ()
() ()
Let | | ()
()
Minor of | | ()
()
Minor of | | ()

Minor of | |

Similarly we can obtain the minor of all other elements.

Cofactor:- If is the minor of any element of a

determinant then its cofactor is given by ( ) .

Let | |

Cofactor of () || Note If | | be a determinant and ,

Minor of () || are cofactor of the elements of
then it can be expanded as
Minor of () ||

Similarly we can obtain the cofactor of all other
elements.

Ex- Find the minor and cofactor of the determinant

||

Soln. Minors are: Assignment-1
|| Q1. Find the value of determinant
||
|| ||
||
|| Q2. Find the value of |
|

|| Q3. If | | | | then find the value of .
||
|| Q4. Find the values of when
||
Cofactors are )| | )| |

Q5. Find the value of and , when

)| | ||

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()

Determinants (सारणिक)

)| | || Proof
|
Q6. Evaluates the followings determinants |

)| | )| | | || || |

Q7. Find the minor and cofactor of the elements of the ( )( )( )
following determinants

Now, | |

)| | )| | )| |

)| | )| | | || || | )
)
 Properties of Determinants ( ) ( )( )+
1. The value of a determinant remains unchanged when ( )( )(

rows are interchange into corresponding *( ) ( ) (
elements.(किसी सारणिि िे पकं ्तियो िो सगं ि स्िभं ों
Proved
मंे बदऱने पर सारणिि िा मान नही बदऱिा है|)

Proof: | 3. If any two rows or two columns are identical then the
| value of determinant is equal to zero. (यदद किसी
सारणिि िा िोई दो पकं ्ति या िोई दो स्िंभ समान हो
| || || | ) िो सारणिि िा मान शुन्य होिा है|)
( )( )(
Proof

||

Now, | |

| || || | | || || | )
( )( )( ( )( )(

)

( ) ( )( ) 4. If each element of any row or any column is multiplied
by a constant quantity ‘k’ then the value of resultant
2. If any two rows or two columns of a determinant are determinant became ‘k’ times of its original. (यदद किसी
interchanged then the determinant retains its absolute सारणिि िे किसी पंक्ति या स्िंभ िे प्रत्येि अवयव िो
value but changes its sign. (यदद किसी सारणिि िे दो किसी र्नयि संख्या ‘k’ से गुिा िर ददया जाय िो
पंक्तियों या दो स्िभं ों िो पररवर्ििि िर किया जािा है, पररिामी सारणिि िा मान मूऱ सारणिि िा k गुिा हो
िो सारणिि िा मान अपररवर्ििि रहिा है परंिु उसिा जािा है|)
चिन्ह बदऱ जािा है|)
| || |

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()

Determinants (सारणिक)

Proof | | | |* +
|
|= | | | | | |
Now,
| Note

( )( )( ) 1. If and then

represents interchange of and .

2. We represents interchange of rows by and of

columns by .

|= | | || 3. If the corresponding elements of two rows or two

columns of a determinants are proportional i.e, in the

same ratio then its value is zero.

|| 4. If all the elements of a row or a column of a
*( ) ( ) (
determinant are zero then its value is also equal to

)+ zero.

| || | Assignment-2
|
5. If some or all elements of a row or a column of a Q1. Evaluates |
determinant are expressed as sum of two or more Q2.Prove that| | ( )( )( )
terms , then the determinant can be expressed as sum Q3. Prove that | |
of two or more determinants. (यदद किसी सारणिि िी Q4. Show that |
|
एि पकं ्ति या स्िंभ िे िु छ या सभी अवयव दो या दो
से अचिि पदों िे योगफऱ िे रूप मंे व्यति हों िो

सारणिि िो दो या दो से अचिि सारणििों िे योगफऱ

िे रूप में व्यति किया जा सििा है|)

| | | || | Q5. Find the value of | |

Proof |( )| | Q6. Prove that
L.H.S | ||

( )| | ( )| |

{ | | | | | |}+{ | | Q7. Prove that | |

| | | |} Q8. Prove that | |
| | | | =R.H.S

6. The value of a determinant does not change when any Q9. Prove that

row column is multiplied by a number of an expression

and is added to or subtracted from any other row or | | ( )( )( )( )

column. (यदद किसी सारणिि िे किसी पकं ्ति या किसी

स्िभं िो किसी सखं ्या या व्यंजि से गुिा िर किसी Q10. Evaluates | |
दसु रे पंक्ति या स्िंभ में जोड़ा या से घटाया जाय िो Q11. Show that
सारणिि िा मान नहीं बदऱिा है|)

( ) सपं कक सूत्र

()

Determinants (सारणिक)

| |( ) ( ) ) | ( )
| ( ( ) (

Hints: ; Hints: * )+ |
)
)
Q12. Find the value of | |

Q24. Prove that | ||

Q13. Prove that | | ( )( )( ) Hints: *

+

Q14. Prove that | | Q25. If then prove that

are different and | | then prove ( )( ) ( )( )(
) |( ) ( )| )
( )(
Q15. If )

that . Hints: Let

Q16. Show that Then ()
Also, )( )
| |( ( )( ) (
(

Q17. Prove that | () Similarly

( )| ( )( )
() |( ) ( )|
( )(
Q18. Prove that | |( ) )

()

Q19. Prove that | () |
|
()

|( ) Now proceed as Q23.

Q20. Prove that Q26. Prove that

| |( ) | |

Q21. If are positive and different then prove that the Q27. Prove that

determinant | | is negative. | | ( )( )( )( )
)
Q22. Prove that Q29. Prove that
| | ( )( )( )
()
| ( )|

(

Q23. Prove that |. Q30. Show that
|
()
| () | (
)
Q24. Prove that (

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()

Determinants (सारणिक)

Hints: Apply take common a. ( ) ( ) ( )
from
respectively. b. ( ) ( ) ( )

c. ( ) ( ) ( )

Q31. Prove that

Q3. Find such that the following points are collinear
( ) ( ) ( ).

|| Q4. If the points ( ) ( ) ( ) are collinear then prove
|| that .

Hints: Apply 2. Solving a System of Linear Equation Using Determinant
() a. System of Linear Equation in Two Variables

Q32. If are positive and are the pth, qth and rth terms Let -------(1)
-------(2)
respectively of a G.P then show that
be two linear equations in and then

||

Where, | || | and

Q33. If are positive and unequal then show that the ||

value of is negative where

|| 1. If then the system is consistent and has a

 Application of Determinants unique solution.

2. If then the system has infinite

1. Area of a triangle in determinant form number of solution.
Let ( ) ( ) and ( ) are the vertices of a
3. If and or then the system is
triangle then area of is given as
inconsistent and have no solution.
* ( ) ( ) ( )+

|| b. System of Linear Equation in Three Variables
Let ( )
()

Note and ( )

a. As area of triangle is positive so se take only absolute be three linear equations in then
value of .
Where, | |
b. If the points ( ) ( ) and ( ) are
collinear points then | || | and

| || |

Assignment-3 ||

Q1. Find the areas of the triangles whose vertices are

a. ( ) ( )( )

b. ( ) ( ) ( ) 1. If then the system is consistence and has

c. ( ) ( ) ( ) unique solution.

d. ( ) ( ) ( ) 2. If and any of is non-zero

e. ( ( )) ( ( )) and then the system is inconsistence and has no

( ( )) solution.

Q2. Using determinant show that the followings points are 3. If then the system is
collinear
consistence and has infinitely many solutions.

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()

Determinants (सारणिक)

Note

1. System of three linear equation in two variables

} is consistence if | |

Also in this case the number of solutions is one.
2. System of homogeneous linear equations in three

variables say

} has no trivial solution if

||

In case of non-trivial solution, number of solution is
infinite.

Assignments-4

Q1. Solve the following equations by Cramer’s Rule

a. and
b.
c. and
d. and
e.

f.

g.

h.

Q2. Check whether the following system of equations are
consistence or not.

a.

b.

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