Solutions of System of Linear Equations | Mathematics Optional Notes for UPSC PDF Download

 Page 1


Edurev123 
2. Solutions of System of Linear Equations 
2.1 The equation ?? ?? +???? +?? =?? has two real roots ?? and ?? . Show that the 
iterative method given by: ?? ?? +?? =-
(?? ?? ?? +?? )
?? ?? ,?? =?? ,?? ,?? ….. is convergent near ?? =?? , 
if |?? |>|?? |. 
(2009 : 6 Marks) 
Solution: 
Let the iterations are given by 
?? ?? +1
=-
(?? ?? ?? +?? )
?? ?? =?? (?? ?? ) (say) (??) 
Now by the known theorem, if ?? (?? ) and ?? '
(?? ) are continuous in an interval about a root a 
of the equation ?? =?? (?? ) and if |?? '
(?? )|<1 for all ?? in the internal, then the successive 
approximations ?? 1
,?? 2
,…. are given by ?? ?? =?? (?? ?? -1
)(?? =1,2,…) converges to the root ?? 
provided that the initial approximation ?? 0
 is chosen in the interval. 
? The iterations converges to ?? if 
|?? (?? )|=|
-?? ?? ?? 2
|<1 ???????? (??) 
Note that ?? '
(?? ) is continuous near ' ?? ' if the iteration converges to ?? =?? . So, we require 
|?? '
(?? )|?=|
?? ?? 2
|<1 
|?? |<|?? |
2
?????????????????????????????????????????????????????????????????????????(???? ) 
Now, given that ?? and ?? are roots of the equation 
?????????????????????????????????? 2
+???? +?? =0
                                   So, ??? +?? =-?? ????????????????????????????????????????????????????? =?? ????????????????????????????????????????????????????????????????????????????????????????????(?????? )
 From (ii) and (iii) 
?????????????????????????????????????????????|???? |<|?? |
2
??????????????????????????|?? |(|?? |-|?? |)>0
???????????????????????????????????????????????|?? |>0
 
2.2 Given the system of equations 
Page 2


Edurev123 
2. Solutions of System of Linear Equations 
2.1 The equation ?? ?? +???? +?? =?? has two real roots ?? and ?? . Show that the 
iterative method given by: ?? ?? +?? =-
(?? ?? ?? +?? )
?? ?? ,?? =?? ,?? ,?? ….. is convergent near ?? =?? , 
if |?? |>|?? |. 
(2009 : 6 Marks) 
Solution: 
Let the iterations are given by 
?? ?? +1
=-
(?? ?? ?? +?? )
?? ?? =?? (?? ?? ) (say) (??) 
Now by the known theorem, if ?? (?? ) and ?? '
(?? ) are continuous in an interval about a root a 
of the equation ?? =?? (?? ) and if |?? '
(?? )|<1 for all ?? in the internal, then the successive 
approximations ?? 1
,?? 2
,…. are given by ?? ?? =?? (?? ?? -1
)(?? =1,2,…) converges to the root ?? 
provided that the initial approximation ?? 0
 is chosen in the interval. 
? The iterations converges to ?? if 
|?? (?? )|=|
-?? ?? ?? 2
|<1 ???????? (??) 
Note that ?? '
(?? ) is continuous near ' ?? ' if the iteration converges to ?? =?? . So, we require 
|?? '
(?? )|?=|
?? ?? 2
|<1 
|?? |<|?? |
2
?????????????????????????????????????????????????????????????????????????(???? ) 
Now, given that ?? and ?? are roots of the equation 
?????????????????????????????????? 2
+???? +?? =0
                                   So, ??? +?? =-?? ????????????????????????????????????????????????????? =?? ????????????????????????????????????????????????????????????????????????????????????????????(?????? )
 From (ii) and (iii) 
?????????????????????????????????????????????|???? |<|?? |
2
??????????????????????????|?? |(|?? |-|?? |)>0
???????????????????????????????????????????????|?? |>0
 
2.2 Given the system of equations 
?? ?? +?? ?? =?? ?? ?? +?? ?? +?? =?? ?? ?? +?? ?? +???? =?? ???? +???? =?? 
State the solubility ?? uniqueness conditions for the system. Give the solution 
when it exists. 
(2010 : 20 Marks) 
Solution: 
From numerical analysis and computer programming. 
The given system of equation can be written as 
[
2 3 0 0 1
2 4 1 0 2
0 2 6 ?? 4
0 0 4 ?? ?? ] 
Applying ?? 2
??? 2
-?? 1
, we get 
 
[
2 3 0 0 1
0 1 1 0 1
0 2 6 ?? 4
0 0 4 ?? ?? ]
[
2 3 0 0 1
0 1 1 0 1
0 0 4 ?? 2
0 0 4 ?? ?? ]
[
2 3 0 0 1
0 1 1 0 1
0 0 4 ?? 2
0 0 0 ?? -?? ?? -2
]
 
Unique solution if 
?? -?? ?=0 and ?? -2?0
?? ???? and ?? ?2
 
Infinite solution if 
?? -?? ?=0 or ?? =?? ?? -2?=0 or ?? =2
?? ?=?? and ?? ?2 or ?? ??? and ?? =2
 
Unique solution: 
Page 3


Edurev123 
2. Solutions of System of Linear Equations 
2.1 The equation ?? ?? +???? +?? =?? has two real roots ?? and ?? . Show that the 
iterative method given by: ?? ?? +?? =-
(?? ?? ?? +?? )
?? ?? ,?? =?? ,?? ,?? ….. is convergent near ?? =?? , 
if |?? |>|?? |. 
(2009 : 6 Marks) 
Solution: 
Let the iterations are given by 
?? ?? +1
=-
(?? ?? ?? +?? )
?? ?? =?? (?? ?? ) (say) (??) 
Now by the known theorem, if ?? (?? ) and ?? '
(?? ) are continuous in an interval about a root a 
of the equation ?? =?? (?? ) and if |?? '
(?? )|<1 for all ?? in the internal, then the successive 
approximations ?? 1
,?? 2
,…. are given by ?? ?? =?? (?? ?? -1
)(?? =1,2,…) converges to the root ?? 
provided that the initial approximation ?? 0
 is chosen in the interval. 
? The iterations converges to ?? if 
|?? (?? )|=|
-?? ?? ?? 2
|<1 ???????? (??) 
Note that ?? '
(?? ) is continuous near ' ?? ' if the iteration converges to ?? =?? . So, we require 
|?? '
(?? )|?=|
?? ?? 2
|<1 
|?? |<|?? |
2
?????????????????????????????????????????????????????????????????????????(???? ) 
Now, given that ?? and ?? are roots of the equation 
?????????????????????????????????? 2
+???? +?? =0
                                   So, ??? +?? =-?? ????????????????????????????????????????????????????? =?? ????????????????????????????????????????????????????????????????????????????????????????????(?????? )
 From (ii) and (iii) 
?????????????????????????????????????????????|???? |<|?? |
2
??????????????????????????|?? |(|?? |-|?? |)>0
???????????????????????????????????????????????|?? |>0
 
2.2 Given the system of equations 
?? ?? +?? ?? =?? ?? ?? +?? ?? +?? =?? ?? ?? +?? ?? +???? =?? ???? +???? =?? 
State the solubility ?? uniqueness conditions for the system. Give the solution 
when it exists. 
(2010 : 20 Marks) 
Solution: 
From numerical analysis and computer programming. 
The given system of equation can be written as 
[
2 3 0 0 1
2 4 1 0 2
0 2 6 ?? 4
0 0 4 ?? ?? ] 
Applying ?? 2
??? 2
-?? 1
, we get 
 
[
2 3 0 0 1
0 1 1 0 1
0 2 6 ?? 4
0 0 4 ?? ?? ]
[
2 3 0 0 1
0 1 1 0 1
0 0 4 ?? 2
0 0 4 ?? ?? ]
[
2 3 0 0 1
0 1 1 0 1
0 0 4 ?? 2
0 0 0 ?? -?? ?? -2
]
 
Unique solution if 
?? -?? ?=0 and ?? -2?0
?? ???? and ?? ?2
 
Infinite solution if 
?? -?? ?=0 or ?? =?? ?? -2?=0 or ?? =2
?? ?=?? and ?? ?2 or ?? ??? and ?? =2
 
Unique solution: 
(?? -?? )?? ?=?? -2
?? ??=
?? -2
?? -?? 4?? +???? ?=2
???? ?=
2-
?? (?? -2)
?? -?? 4
=
2?? -????
4(?? -?? )
?? +?? ?=1??? =1-?? =1-
2?? -????
4(?? -?? )
=
2?? -4?? +????
4(?? -?? )
2?? +3?? ?=1??? =
1
2
(1-3?? )
?? ?=
1
2
{1-3×
(2?? -4?? +???? )
4(?? -?? )
}=
1
2
(8?? -2?? -3???? )
4(?? -?? )
???? ?=
1
2
(8?? -2?? -3???? )
4(?? -?? )
,?? =
2?? -4?? +????
4(?? -?? )
?? ?=
2?? -????
4(?? -?? )
,?? =
?? -2
?? -?? 
Many solutions: 
?? ?=?? ,?? =2
4?? +???? ?=2??? =
2-????
4
?? ?=1-?? =1-(
2-????
4
)??? =
2+????
4
2?? ?=1-3?? ??? =(1-
6+3????
4
)??? =
1
2
(-2-3???? )
4
 
2.3 Solve the following system of simultaneous equations, using Gauss-Seidel 
iterative method : 
?? ?? +???? ?? -?? ?=-????
???? ?? +?? -?? ?? ?=????
?? ?? -?? ?? +???? ?? ?=????
 
(2012 : 20 Marks) 
Solution: 
Solving each equation for the unknown having the largest coefficient, the new equations 
are : 
?? =
1
20
(17-?? +2?? ) (??)
?? =
1
20
(-18-3?? +?? ) (???? )
?? =
1
20
(25-2?? +3?? ) (?????? )
 
Page 4


Edurev123 
2. Solutions of System of Linear Equations 
2.1 The equation ?? ?? +???? +?? =?? has two real roots ?? and ?? . Show that the 
iterative method given by: ?? ?? +?? =-
(?? ?? ?? +?? )
?? ?? ,?? =?? ,?? ,?? ….. is convergent near ?? =?? , 
if |?? |>|?? |. 
(2009 : 6 Marks) 
Solution: 
Let the iterations are given by 
?? ?? +1
=-
(?? ?? ?? +?? )
?? ?? =?? (?? ?? ) (say) (??) 
Now by the known theorem, if ?? (?? ) and ?? '
(?? ) are continuous in an interval about a root a 
of the equation ?? =?? (?? ) and if |?? '
(?? )|<1 for all ?? in the internal, then the successive 
approximations ?? 1
,?? 2
,…. are given by ?? ?? =?? (?? ?? -1
)(?? =1,2,…) converges to the root ?? 
provided that the initial approximation ?? 0
 is chosen in the interval. 
? The iterations converges to ?? if 
|?? (?? )|=|
-?? ?? ?? 2
|<1 ???????? (??) 
Note that ?? '
(?? ) is continuous near ' ?? ' if the iteration converges to ?? =?? . So, we require 
|?? '
(?? )|?=|
?? ?? 2
|<1 
|?? |<|?? |
2
?????????????????????????????????????????????????????????????????????????(???? ) 
Now, given that ?? and ?? are roots of the equation 
?????????????????????????????????? 2
+???? +?? =0
                                   So, ??? +?? =-?? ????????????????????????????????????????????????????? =?? ????????????????????????????????????????????????????????????????????????????????????????????(?????? )
 From (ii) and (iii) 
?????????????????????????????????????????????|???? |<|?? |
2
??????????????????????????|?? |(|?? |-|?? |)>0
???????????????????????????????????????????????|?? |>0
 
2.2 Given the system of equations 
?? ?? +?? ?? =?? ?? ?? +?? ?? +?? =?? ?? ?? +?? ?? +???? =?? ???? +???? =?? 
State the solubility ?? uniqueness conditions for the system. Give the solution 
when it exists. 
(2010 : 20 Marks) 
Solution: 
From numerical analysis and computer programming. 
The given system of equation can be written as 
[
2 3 0 0 1
2 4 1 0 2
0 2 6 ?? 4
0 0 4 ?? ?? ] 
Applying ?? 2
??? 2
-?? 1
, we get 
 
[
2 3 0 0 1
0 1 1 0 1
0 2 6 ?? 4
0 0 4 ?? ?? ]
[
2 3 0 0 1
0 1 1 0 1
0 0 4 ?? 2
0 0 4 ?? ?? ]
[
2 3 0 0 1
0 1 1 0 1
0 0 4 ?? 2
0 0 0 ?? -?? ?? -2
]
 
Unique solution if 
?? -?? ?=0 and ?? -2?0
?? ???? and ?? ?2
 
Infinite solution if 
?? -?? ?=0 or ?? =?? ?? -2?=0 or ?? =2
?? ?=?? and ?? ?2 or ?? ??? and ?? =2
 
Unique solution: 
(?? -?? )?? ?=?? -2
?? ??=
?? -2
?? -?? 4?? +???? ?=2
???? ?=
2-
?? (?? -2)
?? -?? 4
=
2?? -????
4(?? -?? )
?? +?? ?=1??? =1-?? =1-
2?? -????
4(?? -?? )
=
2?? -4?? +????
4(?? -?? )
2?? +3?? ?=1??? =
1
2
(1-3?? )
?? ?=
1
2
{1-3×
(2?? -4?? +???? )
4(?? -?? )
}=
1
2
(8?? -2?? -3???? )
4(?? -?? )
???? ?=
1
2
(8?? -2?? -3???? )
4(?? -?? )
,?? =
2?? -4?? +????
4(?? -?? )
?? ?=
2?? -????
4(?? -?? )
,?? =
?? -2
?? -?? 
Many solutions: 
?? ?=?? ,?? =2
4?? +???? ?=2??? =
2-????
4
?? ?=1-?? =1-(
2-????
4
)??? =
2+????
4
2?? ?=1-3?? ??? =(1-
6+3????
4
)??? =
1
2
(-2-3???? )
4
 
2.3 Solve the following system of simultaneous equations, using Gauss-Seidel 
iterative method : 
?? ?? +???? ?? -?? ?=-????
???? ?? +?? -?? ?? ?=????
?? ?? -?? ?? +???? ?? ?=????
 
(2012 : 20 Marks) 
Solution: 
Solving each equation for the unknown having the largest coefficient, the new equations 
are : 
?? =
1
20
(17-?? +2?? ) (??)
?? =
1
20
(-18-3?? +?? ) (???? )
?? =
1
20
(25-2?? +3?? ) (?????? )
 
Choosing ?? =0,?? =0 to start with, we get, 
?? (1)
=
1
20
(17-0+0)=0.85 (first approximation)  
Putting ?? =?? (1)
=0.85,?? =0 in (ii) 
?? (1)
=
1
20
(-18-2.55)=-1.0275 
Putting ?? =?? (1)
=0.85,?? =?? (1)
=-1.0275 in (iii), we get 
?? (1)
=
1
20
(25-1.7-3.0825)=1.0109
?? (2)
=
1
20
(17+1.0275+2.0218)=1.0025
?? (2)
=
1
20
(-18-3.0075+1.0109)=-0.9998
?? (2)
=
1
20
(25-2.0050-2.9994)=0.9998
?? (3)
=
1
20
(17+0.9998+1.9996)=0.99997
?? (3)
=
1
20
(-18-2.99991+0.9998)=-1.0000
?? (3)
=
1
20
(25-1.99994-3.0000)=1.0000
?? (4)
=
1
20
(17+1.0000+2.0000)=1.0000
 
?? (4)
=
1
20
(-18-3.0000+1.0000)=-1.0000
?? (4)
=
1
20
(25-2.0000-3.0000)=1.0000
 
Since ?? (4)
,?? (4)
,?? (4)
 are sufficiently close to ?? (3)
,?? (3)
,?? (3)
 respectively, so that values 
1.0000,-1.0000,1.0000 can be taken as the solution of the given system. 
2.4 Solve the system of equations : 
?? ?? ?? -?? ?? =?? -?? ?? +?? ?? ?? -?? ?? =?? -?? ?? +?? ?? ?? =?? 
Using Gauss-Seidel iteration method (Perform three iterations). 
(2014 : 15 Marks) 
Solution: 
The given system of equations can be written as 
Page 5


Edurev123 
2. Solutions of System of Linear Equations 
2.1 The equation ?? ?? +???? +?? =?? has two real roots ?? and ?? . Show that the 
iterative method given by: ?? ?? +?? =-
(?? ?? ?? +?? )
?? ?? ,?? =?? ,?? ,?? ….. is convergent near ?? =?? , 
if |?? |>|?? |. 
(2009 : 6 Marks) 
Solution: 
Let the iterations are given by 
?? ?? +1
=-
(?? ?? ?? +?? )
?? ?? =?? (?? ?? ) (say) (??) 
Now by the known theorem, if ?? (?? ) and ?? '
(?? ) are continuous in an interval about a root a 
of the equation ?? =?? (?? ) and if |?? '
(?? )|<1 for all ?? in the internal, then the successive 
approximations ?? 1
,?? 2
,…. are given by ?? ?? =?? (?? ?? -1
)(?? =1,2,…) converges to the root ?? 
provided that the initial approximation ?? 0
 is chosen in the interval. 
? The iterations converges to ?? if 
|?? (?? )|=|
-?? ?? ?? 2
|<1 ???????? (??) 
Note that ?? '
(?? ) is continuous near ' ?? ' if the iteration converges to ?? =?? . So, we require 
|?? '
(?? )|?=|
?? ?? 2
|<1 
|?? |<|?? |
2
?????????????????????????????????????????????????????????????????????????(???? ) 
Now, given that ?? and ?? are roots of the equation 
?????????????????????????????????? 2
+???? +?? =0
                                   So, ??? +?? =-?? ????????????????????????????????????????????????????? =?? ????????????????????????????????????????????????????????????????????????????????????????????(?????? )
 From (ii) and (iii) 
?????????????????????????????????????????????|???? |<|?? |
2
??????????????????????????|?? |(|?? |-|?? |)>0
???????????????????????????????????????????????|?? |>0
 
2.2 Given the system of equations 
?? ?? +?? ?? =?? ?? ?? +?? ?? +?? =?? ?? ?? +?? ?? +???? =?? ???? +???? =?? 
State the solubility ?? uniqueness conditions for the system. Give the solution 
when it exists. 
(2010 : 20 Marks) 
Solution: 
From numerical analysis and computer programming. 
The given system of equation can be written as 
[
2 3 0 0 1
2 4 1 0 2
0 2 6 ?? 4
0 0 4 ?? ?? ] 
Applying ?? 2
??? 2
-?? 1
, we get 
 
[
2 3 0 0 1
0 1 1 0 1
0 2 6 ?? 4
0 0 4 ?? ?? ]
[
2 3 0 0 1
0 1 1 0 1
0 0 4 ?? 2
0 0 4 ?? ?? ]
[
2 3 0 0 1
0 1 1 0 1
0 0 4 ?? 2
0 0 0 ?? -?? ?? -2
]
 
Unique solution if 
?? -?? ?=0 and ?? -2?0
?? ???? and ?? ?2
 
Infinite solution if 
?? -?? ?=0 or ?? =?? ?? -2?=0 or ?? =2
?? ?=?? and ?? ?2 or ?? ??? and ?? =2
 
Unique solution: 
(?? -?? )?? ?=?? -2
?? ??=
?? -2
?? -?? 4?? +???? ?=2
???? ?=
2-
?? (?? -2)
?? -?? 4
=
2?? -????
4(?? -?? )
?? +?? ?=1??? =1-?? =1-
2?? -????
4(?? -?? )
=
2?? -4?? +????
4(?? -?? )
2?? +3?? ?=1??? =
1
2
(1-3?? )
?? ?=
1
2
{1-3×
(2?? -4?? +???? )
4(?? -?? )
}=
1
2
(8?? -2?? -3???? )
4(?? -?? )
???? ?=
1
2
(8?? -2?? -3???? )
4(?? -?? )
,?? =
2?? -4?? +????
4(?? -?? )
?? ?=
2?? -????
4(?? -?? )
,?? =
?? -2
?? -?? 
Many solutions: 
?? ?=?? ,?? =2
4?? +???? ?=2??? =
2-????
4
?? ?=1-?? =1-(
2-????
4
)??? =
2+????
4
2?? ?=1-3?? ??? =(1-
6+3????
4
)??? =
1
2
(-2-3???? )
4
 
2.3 Solve the following system of simultaneous equations, using Gauss-Seidel 
iterative method : 
?? ?? +???? ?? -?? ?=-????
???? ?? +?? -?? ?? ?=????
?? ?? -?? ?? +???? ?? ?=????
 
(2012 : 20 Marks) 
Solution: 
Solving each equation for the unknown having the largest coefficient, the new equations 
are : 
?? =
1
20
(17-?? +2?? ) (??)
?? =
1
20
(-18-3?? +?? ) (???? )
?? =
1
20
(25-2?? +3?? ) (?????? )
 
Choosing ?? =0,?? =0 to start with, we get, 
?? (1)
=
1
20
(17-0+0)=0.85 (first approximation)  
Putting ?? =?? (1)
=0.85,?? =0 in (ii) 
?? (1)
=
1
20
(-18-2.55)=-1.0275 
Putting ?? =?? (1)
=0.85,?? =?? (1)
=-1.0275 in (iii), we get 
?? (1)
=
1
20
(25-1.7-3.0825)=1.0109
?? (2)
=
1
20
(17+1.0275+2.0218)=1.0025
?? (2)
=
1
20
(-18-3.0075+1.0109)=-0.9998
?? (2)
=
1
20
(25-2.0050-2.9994)=0.9998
?? (3)
=
1
20
(17+0.9998+1.9996)=0.99997
?? (3)
=
1
20
(-18-2.99991+0.9998)=-1.0000
?? (3)
=
1
20
(25-1.99994-3.0000)=1.0000
?? (4)
=
1
20
(17+1.0000+2.0000)=1.0000
 
?? (4)
=
1
20
(-18-3.0000+1.0000)=-1.0000
?? (4)
=
1
20
(25-2.0000-3.0000)=1.0000
 
Since ?? (4)
,?? (4)
,?? (4)
 are sufficiently close to ?? (3)
,?? (3)
,?? (3)
 respectively, so that values 
1.0000,-1.0000,1.0000 can be taken as the solution of the given system. 
2.4 Solve the system of equations : 
?? ?? ?? -?? ?? =?? -?? ?? +?? ?? ?? -?? ?? =?? -?? ?? +?? ?? ?? =?? 
Using Gauss-Seidel iteration method (Perform three iterations). 
(2014 : 15 Marks) 
Solution: 
The given system of equations can be written as 
?? 1
=
1
2
(7+?? 2
)
?? 2
=
1
2
(1+?? 1
+?? 3
)
?? 3
=
1
2
(1+?? 2
)
}
 
 
 
 
(??) 
By the Gauss-Seidel method, system (i) can be written as 
?? 1
?? +1
=
1
2
(7+?? 2
(?? )
)
?? 2
?? +1
=
1
2
(1+?? 2
(?? +1)
+?? 3
?? )
?? 3
?? +1
=
1
2
(1+?? 2
(?? +1)
)
 
where, ?? =0,1,2,3… 
Now taking,??? (0)=0, we obtain the following iterations 
?? =0 : 
?? ?
(1)
=
1
2
(7+0)=
7
2
=3.5 
?? 2
(1)
=
1
2
(1+3.5+0)=
4.5
2
=2.25 
?? 3
(1)
=
1
2
(1+2.25)=
1
2
(3.25)=1.625 
?? =1?: 
?? 1
(2)
=
1
2
(7+?? 2
(1)
)=
1
2
(7+2.25)=
9.25
2
=4.625 
?? 2
(2)
=
1
2
(1+?? 1
(2)
+?? 3
(1)
)=
1
2
(1+4.625+1.625)=3.625 
?? 3
(2)
=
1
2
(1+?? 2
(2)
)=
1
2
(1+4.625+1.625)=3.625 
?? 3
(2)
=
1
2
(1+?? 2
(2)
)=
1
2
(1+3.625)=2.3125 
?? =2 : 
?? 1
(3)
=
1
2
(7+?? 2
(2)
)=
1
2
(7+3.625)=5.3125 
?? 2
(3)
=
1
2
(1+?? 1
(4)
+?? 3
(2)
)=
1
2
(1+5.3125+2.3125)=4.3125