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Page 1
Edurev123
3 Matrix
3.1 Find a hermitian and a skew-hermitian matrix each whose sum is the matrix
[
?? ?? ?? -?? ?? ?? +?? ?? ?? -?? +?? ?? ?? ?? ]
(2009 : 12 Marks)
Solution:
Given any matrix ?? we can write it as
?? =
1
2
(?? +?? ?
)+
1
2
(?? -?? *
)
where ?? *
is the complex tranjugate of ?? . Also
1
2
(?? +?? *
) is always hermitian as
[
1
2
(?? +?? *
)]
*
=
1
2
(?? +?? *
)
*
=
1
2
(?? *
+?? )
=
1
2
(?? +?? *
)
And
1
2
(?? +?? *
) is skew hermitian as
[
1
2
(?? -?? *
)]* =
1
2
(?? *
-?? )=-[
1
2
(?? -?? *
)]
? ?? =
1
2
[?? +?? *
]+
1
2
[?? -?? *
]
=?? +??
where ?? is hermitian and ?? skew hermitian.
Taking ?? as given matrix.
?? =
1
2
(?? +?? *
)
=
1
2
{[
2?? 3 -1
1 2+3?? 2
-?? +1 4 5?? ]+[
-2?? ?? ?? +1
3 2-3?? 4
-1 2 -5?? ]}
=[
0 2 ??/2
2 2 3
-??/2 3 0
] which is hermitian.
?????? ?? =
1
2
(?? -?? *
)
Page 2
Edurev123
3 Matrix
3.1 Find a hermitian and a skew-hermitian matrix each whose sum is the matrix
[
?? ?? ?? -?? ?? ?? +?? ?? ?? -?? +?? ?? ?? ?? ]
(2009 : 12 Marks)
Solution:
Given any matrix ?? we can write it as
?? =
1
2
(?? +?? ?
)+
1
2
(?? -?? *
)
where ?? *
is the complex tranjugate of ?? . Also
1
2
(?? +?? *
) is always hermitian as
[
1
2
(?? +?? *
)]
*
=
1
2
(?? +?? *
)
*
=
1
2
(?? *
+?? )
=
1
2
(?? +?? *
)
And
1
2
(?? +?? *
) is skew hermitian as
[
1
2
(?? -?? *
)]* =
1
2
(?? *
-?? )=-[
1
2
(?? -?? *
)]
? ?? =
1
2
[?? +?? *
]+
1
2
[?? -?? *
]
=?? +??
where ?? is hermitian and ?? skew hermitian.
Taking ?? as given matrix.
?? =
1
2
(?? +?? *
)
=
1
2
{[
2?? 3 -1
1 2+3?? 2
-?? +1 4 5?? ]+[
-2?? ?? ?? +1
3 2-3?? 4
-1 2 -5?? ]}
=[
0 2 ??/2
2 2 3
-??/2 3 0
] which is hermitian.
?????? ?? =
1
2
(?? -?? *
)
=
1
2
{[
2?? 3 -1
1 2+3?? 2
-?? +1 4 5?? ]-[
-2?? ?? ?? +1
3 2-3?? 4
-1 2 -5?? ]}
=
[
2?? 1
-(?? +2)
2
-1 3?? -1
(-?? +2)
2
1 5?? ]
So, ?? and ?? are required vector where
?? =?? +??
and ?? is Hermitian and ?? skew Hermitian.
3.2 Find a ?? ×?? real matrix ?? which is both orthogonal and skew symmetric. Can
there exist a ?? ×?? real matrix which is both orthogonal and skaw symmetric.
Justify your answer.
(2009 : 20 Marks)
Solution:
Approach : Consider the form of a skew symmetric matrix (diagonal elements zero) and
impose conditions for orthogonality.
Let ?? be a 2×2 skew symmetric matrix and ?? =[
?? ?? ?? ?? ].
?? is skew symmetric ? ?? =-?? ??
=> [
?? ?? ?? ?? ]=[
-?? -?? -?? -?? ]??? =??
=0 and ?? =-??
? ?? =[
0 ?? -?? 0
]
If ?? is orthogonal then ?? ?? ?
=??
=> ?? 2
=1??? =±??
?[
0 1
-1 0
] and [
0 -1
1 0
] are the only matrices that are orthogonal and skew symmetric.
Again let ?? be a 3×3 skew symmetric matrix. Then
?? =[
0 ?? ?? -?? 0 ?? -?? -?? 0
] as seen in previous case.
Also if ?? is orthogonal.
Page 3
Edurev123
3 Matrix
3.1 Find a hermitian and a skew-hermitian matrix each whose sum is the matrix
[
?? ?? ?? -?? ?? ?? +?? ?? ?? -?? +?? ?? ?? ?? ]
(2009 : 12 Marks)
Solution:
Given any matrix ?? we can write it as
?? =
1
2
(?? +?? ?
)+
1
2
(?? -?? *
)
where ?? *
is the complex tranjugate of ?? . Also
1
2
(?? +?? *
) is always hermitian as
[
1
2
(?? +?? *
)]
*
=
1
2
(?? +?? *
)
*
=
1
2
(?? *
+?? )
=
1
2
(?? +?? *
)
And
1
2
(?? +?? *
) is skew hermitian as
[
1
2
(?? -?? *
)]* =
1
2
(?? *
-?? )=-[
1
2
(?? -?? *
)]
? ?? =
1
2
[?? +?? *
]+
1
2
[?? -?? *
]
=?? +??
where ?? is hermitian and ?? skew hermitian.
Taking ?? as given matrix.
?? =
1
2
(?? +?? *
)
=
1
2
{[
2?? 3 -1
1 2+3?? 2
-?? +1 4 5?? ]+[
-2?? ?? ?? +1
3 2-3?? 4
-1 2 -5?? ]}
=[
0 2 ??/2
2 2 3
-??/2 3 0
] which is hermitian.
?????? ?? =
1
2
(?? -?? *
)
=
1
2
{[
2?? 3 -1
1 2+3?? 2
-?? +1 4 5?? ]-[
-2?? ?? ?? +1
3 2-3?? 4
-1 2 -5?? ]}
=
[
2?? 1
-(?? +2)
2
-1 3?? -1
(-?? +2)
2
1 5?? ]
So, ?? and ?? are required vector where
?? =?? +??
and ?? is Hermitian and ?? skew Hermitian.
3.2 Find a ?? ×?? real matrix ?? which is both orthogonal and skew symmetric. Can
there exist a ?? ×?? real matrix which is both orthogonal and skaw symmetric.
Justify your answer.
(2009 : 20 Marks)
Solution:
Approach : Consider the form of a skew symmetric matrix (diagonal elements zero) and
impose conditions for orthogonality.
Let ?? be a 2×2 skew symmetric matrix and ?? =[
?? ?? ?? ?? ].
?? is skew symmetric ? ?? =-?? ??
=> [
?? ?? ?? ?? ]=[
-?? -?? -?? -?? ]??? =??
=0 and ?? =-??
? ?? =[
0 ?? -?? 0
]
If ?? is orthogonal then ?? ?? ?
=??
=> ?? 2
=1??? =±??
?[
0 1
-1 0
] and [
0 -1
1 0
] are the only matrices that are orthogonal and skew symmetric.
Again let ?? be a 3×3 skew symmetric matrix. Then
?? =[
0 ?? ?? -?? 0 ?? -?? -?? 0
] as seen in previous case.
Also if ?? is orthogonal.
? ????
?
=?? ? [
0 ?? ?? -?? 0 ?? -?? -?? 0
][
0 -?? -?? ?? 0 -?? ?? ?? 0
] =[
1 0 0
0 1 0
0 0 1
]
? [
?? 2
+?? 2
???? -????
???? ?? 2
+?? 2
????
-???? ???? ?? 2
-?? 2
] =[
1 0 0
0 1 0
0 0 1
]
? ?? 2
+?? 2
=?? 2
+?? 2
=?? 2
+?? 2
=1 …(i)
=???? =???? =???? =0 …(ii)
From (ii) two of ?? ,?? ,?? must be zero if ?? =?? =0??? 2
+?? 2
=0?1.
Similarly in other cases it can be shown the system of equations is not compatible.
So, a 3×3 skew symmetric matrix can not be orthogonal.
3.3. If ?? ?? ,?? ?? ,?? ?? are eigen values of the matrix
?? =[
?? ?? -?? ?? ?? ???? ?? ?? ?? ????
]
Show that v?? ?? ?? +?? ?? ?? +?? ?? ?? =v????????
(2010 : 12 Marks)
Solution:
?????????? ?? h?? ???????????? ?? =[
26 -2 2
2 21 4
4 2 28
]
Now, finding eigen values of ??
Page 4
Edurev123
3 Matrix
3.1 Find a hermitian and a skew-hermitian matrix each whose sum is the matrix
[
?? ?? ?? -?? ?? ?? +?? ?? ?? -?? +?? ?? ?? ?? ]
(2009 : 12 Marks)
Solution:
Given any matrix ?? we can write it as
?? =
1
2
(?? +?? ?
)+
1
2
(?? -?? *
)
where ?? *
is the complex tranjugate of ?? . Also
1
2
(?? +?? *
) is always hermitian as
[
1
2
(?? +?? *
)]
*
=
1
2
(?? +?? *
)
*
=
1
2
(?? *
+?? )
=
1
2
(?? +?? *
)
And
1
2
(?? +?? *
) is skew hermitian as
[
1
2
(?? -?? *
)]* =
1
2
(?? *
-?? )=-[
1
2
(?? -?? *
)]
? ?? =
1
2
[?? +?? *
]+
1
2
[?? -?? *
]
=?? +??
where ?? is hermitian and ?? skew hermitian.
Taking ?? as given matrix.
?? =
1
2
(?? +?? *
)
=
1
2
{[
2?? 3 -1
1 2+3?? 2
-?? +1 4 5?? ]+[
-2?? ?? ?? +1
3 2-3?? 4
-1 2 -5?? ]}
=[
0 2 ??/2
2 2 3
-??/2 3 0
] which is hermitian.
?????? ?? =
1
2
(?? -?? *
)
=
1
2
{[
2?? 3 -1
1 2+3?? 2
-?? +1 4 5?? ]-[
-2?? ?? ?? +1
3 2-3?? 4
-1 2 -5?? ]}
=
[
2?? 1
-(?? +2)
2
-1 3?? -1
(-?? +2)
2
1 5?? ]
So, ?? and ?? are required vector where
?? =?? +??
and ?? is Hermitian and ?? skew Hermitian.
3.2 Find a ?? ×?? real matrix ?? which is both orthogonal and skew symmetric. Can
there exist a ?? ×?? real matrix which is both orthogonal and skaw symmetric.
Justify your answer.
(2009 : 20 Marks)
Solution:
Approach : Consider the form of a skew symmetric matrix (diagonal elements zero) and
impose conditions for orthogonality.
Let ?? be a 2×2 skew symmetric matrix and ?? =[
?? ?? ?? ?? ].
?? is skew symmetric ? ?? =-?? ??
=> [
?? ?? ?? ?? ]=[
-?? -?? -?? -?? ]??? =??
=0 and ?? =-??
? ?? =[
0 ?? -?? 0
]
If ?? is orthogonal then ?? ?? ?
=??
=> ?? 2
=1??? =±??
?[
0 1
-1 0
] and [
0 -1
1 0
] are the only matrices that are orthogonal and skew symmetric.
Again let ?? be a 3×3 skew symmetric matrix. Then
?? =[
0 ?? ?? -?? 0 ?? -?? -?? 0
] as seen in previous case.
Also if ?? is orthogonal.
? ????
?
=?? ? [
0 ?? ?? -?? 0 ?? -?? -?? 0
][
0 -?? -?? ?? 0 -?? ?? ?? 0
] =[
1 0 0
0 1 0
0 0 1
]
? [
?? 2
+?? 2
???? -????
???? ?? 2
+?? 2
????
-???? ???? ?? 2
-?? 2
] =[
1 0 0
0 1 0
0 0 1
]
? ?? 2
+?? 2
=?? 2
+?? 2
=?? 2
+?? 2
=1 …(i)
=???? =???? =???? =0 …(ii)
From (ii) two of ?? ,?? ,?? must be zero if ?? =?? =0??? 2
+?? 2
=0?1.
Similarly in other cases it can be shown the system of equations is not compatible.
So, a 3×3 skew symmetric matrix can not be orthogonal.
3.3. If ?? ?? ,?? ?? ,?? ?? are eigen values of the matrix
?? =[
?? ?? -?? ?? ?? ???? ?? ?? ?? ????
]
Show that v?? ?? ?? +?? ?? ?? +?? ?? ?? =v????????
(2010 : 12 Marks)
Solution:
?????????? ?? h?? ???????????? ?? =[
26 -2 2
2 21 4
4 2 28
]
Now, finding eigen values of ??
[
26-?? -2 2
2 21-?? 4
4 2 28-?? ]=0
? (26-?? )[(21-?? )(28-?? )-8]+2[56-2?? -16]+2[4-84+4?? )=0
? (26-?? )[588-49?? +?? 2
-8]+2[40-2?? ]+2[4?? -80]=0
? (26-?? )[580-49?? +?? 2
]+80-4?? +8?? -160=0
? 15080-1274?? +26?? 2
-580?? +49?? 2
-?? 3
-80+4?? =0
? 15000-1850?? +75?? 2
-?? 3
=0
? ?? 3
-75?? 2
+??850?? -??5000=0
?????? ?? 1
,?? 2
,?? 3
???? ?????????? ???? ???? .(1)
? ?? 1
+?? 2
+?? 3
=75
?? 1
?? 2
+?? 2
?? 3
+?? 3
?? 4
=1850
?? 1
?? 2
?? 3
=15000
???????? , (?? 1
+?? 2
+?? 3
)
2
=?? 2
1
+?? 2
2
+?? 3
2
+2(?? 1
?? 2
+?? 2
?? 3
+?? 3
?? 1
)
Putting above values in above eqn., we get
(75)
2
=?? 1
2
+?? 2
2
+?? 3
2
+2(1850)
? ?? 1
2
+?? 2
2
+?? 3
2
=5625-3700
? ?? 1
2
+?? 2
2
+?? 3
2
=1925=1949
? v?? 1
2
+?? 2
2
+?? 3
2
=v1949
3.4 Let ?? and ?? be ?? ×?? matrices over reals. Show that ?? -???? is invertible if ?? -
???? is invertible. Deduce that ???? and ???? have same eigen values.
(2010 : 20 Marks)
Solution:
Given ?? and ?? are two ?? ×?? matrices over reels.
Let ?? =(?? -???? )
-1
(Assuming that ?? -AB is invertible)
? ?? =?? +???? +(???? )
2
+(???? )
3
+?.. (By binomial expansion)
?????? , ?? ×?? =???? +(???? )
2
+(???? )
3
+?
? ?? ×?? ×?? =?? +???? +(???? )
2
+(???? )
3
+?
=(?? -???? )
-1
?(?? -???? ) is invertible if ?? -???? is invertible.
To show that ???? and ???? have same eigen values. Let ?? be an eigen-value of ???? ,
Case 1 : if ?? =0, then ?????? =0.?? (?? -
?????????? ???????????? )
=> (0.?? -???? )?? =0
Page 5
Edurev123
3 Matrix
3.1 Find a hermitian and a skew-hermitian matrix each whose sum is the matrix
[
?? ?? ?? -?? ?? ?? +?? ?? ?? -?? +?? ?? ?? ?? ]
(2009 : 12 Marks)
Solution:
Given any matrix ?? we can write it as
?? =
1
2
(?? +?? ?
)+
1
2
(?? -?? *
)
where ?? *
is the complex tranjugate of ?? . Also
1
2
(?? +?? *
) is always hermitian as
[
1
2
(?? +?? *
)]
*
=
1
2
(?? +?? *
)
*
=
1
2
(?? *
+?? )
=
1
2
(?? +?? *
)
And
1
2
(?? +?? *
) is skew hermitian as
[
1
2
(?? -?? *
)]* =
1
2
(?? *
-?? )=-[
1
2
(?? -?? *
)]
? ?? =
1
2
[?? +?? *
]+
1
2
[?? -?? *
]
=?? +??
where ?? is hermitian and ?? skew hermitian.
Taking ?? as given matrix.
?? =
1
2
(?? +?? *
)
=
1
2
{[
2?? 3 -1
1 2+3?? 2
-?? +1 4 5?? ]+[
-2?? ?? ?? +1
3 2-3?? 4
-1 2 -5?? ]}
=[
0 2 ??/2
2 2 3
-??/2 3 0
] which is hermitian.
?????? ?? =
1
2
(?? -?? *
)
=
1
2
{[
2?? 3 -1
1 2+3?? 2
-?? +1 4 5?? ]-[
-2?? ?? ?? +1
3 2-3?? 4
-1 2 -5?? ]}
=
[
2?? 1
-(?? +2)
2
-1 3?? -1
(-?? +2)
2
1 5?? ]
So, ?? and ?? are required vector where
?? =?? +??
and ?? is Hermitian and ?? skew Hermitian.
3.2 Find a ?? ×?? real matrix ?? which is both orthogonal and skew symmetric. Can
there exist a ?? ×?? real matrix which is both orthogonal and skaw symmetric.
Justify your answer.
(2009 : 20 Marks)
Solution:
Approach : Consider the form of a skew symmetric matrix (diagonal elements zero) and
impose conditions for orthogonality.
Let ?? be a 2×2 skew symmetric matrix and ?? =[
?? ?? ?? ?? ].
?? is skew symmetric ? ?? =-?? ??
=> [
?? ?? ?? ?? ]=[
-?? -?? -?? -?? ]??? =??
=0 and ?? =-??
? ?? =[
0 ?? -?? 0
]
If ?? is orthogonal then ?? ?? ?
=??
=> ?? 2
=1??? =±??
?[
0 1
-1 0
] and [
0 -1
1 0
] are the only matrices that are orthogonal and skew symmetric.
Again let ?? be a 3×3 skew symmetric matrix. Then
?? =[
0 ?? ?? -?? 0 ?? -?? -?? 0
] as seen in previous case.
Also if ?? is orthogonal.
? ????
?
=?? ? [
0 ?? ?? -?? 0 ?? -?? -?? 0
][
0 -?? -?? ?? 0 -?? ?? ?? 0
] =[
1 0 0
0 1 0
0 0 1
]
? [
?? 2
+?? 2
???? -????
???? ?? 2
+?? 2
????
-???? ???? ?? 2
-?? 2
] =[
1 0 0
0 1 0
0 0 1
]
? ?? 2
+?? 2
=?? 2
+?? 2
=?? 2
+?? 2
=1 …(i)
=???? =???? =???? =0 …(ii)
From (ii) two of ?? ,?? ,?? must be zero if ?? =?? =0??? 2
+?? 2
=0?1.
Similarly in other cases it can be shown the system of equations is not compatible.
So, a 3×3 skew symmetric matrix can not be orthogonal.
3.3. If ?? ?? ,?? ?? ,?? ?? are eigen values of the matrix
?? =[
?? ?? -?? ?? ?? ???? ?? ?? ?? ????
]
Show that v?? ?? ?? +?? ?? ?? +?? ?? ?? =v????????
(2010 : 12 Marks)
Solution:
?????????? ?? h?? ???????????? ?? =[
26 -2 2
2 21 4
4 2 28
]
Now, finding eigen values of ??
[
26-?? -2 2
2 21-?? 4
4 2 28-?? ]=0
? (26-?? )[(21-?? )(28-?? )-8]+2[56-2?? -16]+2[4-84+4?? )=0
? (26-?? )[588-49?? +?? 2
-8]+2[40-2?? ]+2[4?? -80]=0
? (26-?? )[580-49?? +?? 2
]+80-4?? +8?? -160=0
? 15080-1274?? +26?? 2
-580?? +49?? 2
-?? 3
-80+4?? =0
? 15000-1850?? +75?? 2
-?? 3
=0
? ?? 3
-75?? 2
+??850?? -??5000=0
?????? ?? 1
,?? 2
,?? 3
???? ?????????? ???? ???? .(1)
? ?? 1
+?? 2
+?? 3
=75
?? 1
?? 2
+?? 2
?? 3
+?? 3
?? 4
=1850
?? 1
?? 2
?? 3
=15000
???????? , (?? 1
+?? 2
+?? 3
)
2
=?? 2
1
+?? 2
2
+?? 3
2
+2(?? 1
?? 2
+?? 2
?? 3
+?? 3
?? 1
)
Putting above values in above eqn., we get
(75)
2
=?? 1
2
+?? 2
2
+?? 3
2
+2(1850)
? ?? 1
2
+?? 2
2
+?? 3
2
=5625-3700
? ?? 1
2
+?? 2
2
+?? 3
2
=1925=1949
? v?? 1
2
+?? 2
2
+?? 3
2
=v1949
3.4 Let ?? and ?? be ?? ×?? matrices over reals. Show that ?? -???? is invertible if ?? -
???? is invertible. Deduce that ???? and ???? have same eigen values.
(2010 : 20 Marks)
Solution:
Given ?? and ?? are two ?? ×?? matrices over reels.
Let ?? =(?? -???? )
-1
(Assuming that ?? -AB is invertible)
? ?? =?? +???? +(???? )
2
+(???? )
3
+?.. (By binomial expansion)
?????? , ?? ×?? =???? +(???? )
2
+(???? )
3
+?
? ?? ×?? ×?? =?? +???? +(???? )
2
+(???? )
3
+?
=(?? -???? )
-1
?(?? -???? ) is invertible if ?? -???? is invertible.
To show that ???? and ???? have same eigen values. Let ?? be an eigen-value of ???? ,
Case 1 : if ?? =0, then ?????? =0.?? (?? -
?????????? ???????????? )
=> (0.?? -???? )?? =0
? (0.?? -???? )?? =0
? 0 =|0.?? -???? |=|-?? ||?? |=|?? ||-?? |
=|0.?? -???? |
?0 is an eigen-value of ???? aiso.
Case 2 : If ?? ?0. Let ?? is not an eigen-value of ???? . Then
?????? ????? ?(???? -???? )?? ?0
? |???? -???? |?0??? ?? |
1
?? ???? -1| ?0
? |
????
?? -?? |?0?|
?? ?? ·?? -?? |?0
or
|?? -
?? ?? ·?? |?0??? -
?? ?? ·?? is invertihle.
? By abc e deduction, it can be concluded that ?? -?? ·
?? ?? is also invertible.
? |?? -?? ·
?? ?? |?0
? ?? ?? |?? -
?? -?? ?? |?0
? |???? -???? |?0
??? is not an eigen-value of ???? which contradicts our assumption.
??? is an eigen-value of ???? also.
????? and ???? have same eigen-values.
3.5 Let ?? be a non-singular, ?? ×?? square matrix. Show that ?? ( adj. ?? )=|?? |.?? ?? .
Hence, show that ladj. (Adj. ?? )|=|?? |
(?? -?? )
?? .
(2011: 10 Marks)
Solution:
If ?? be any ?? -rowed square matrix, then
(adj.?? )?? =?? (adj.?? )=|?? |?? ?? (??)
where ?? ?? is a unit matrix of order ?? and adj. ?? is adjoint of the matrix ?? .
Replacing ?? by adj. ?? in (i), we get
( adj. A)(adj. adj. A) =|adj.A|?? ??
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Dec 18, 2024 Last updated |
Document Description: Matrix for UPSC 2024 is part of Mathematics Optional Notes for UPSC preparation.
The notes and questions for Matrix have been prepared according to the UPSC exam syllabus. Information about Matrix covers topics
like and Matrix Example, for UPSC 2024 Exam. Find important definitions, questions, notes, meanings, examples, exercises and tests below for Matrix.
Introduction of Matrix in English is available as part of our Mathematics Optional Notes for UPSC
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Full syllabus notes, lecture & questions for Matrix | Mathematics Optional Notes for UPSC - UPSC | Plus excerises question with solution to help you revise complete syllabus for Mathematics Optional Notes for UPSC | Best notes, free PDF download
Information about Matrix
In this doc you can find the meaning of Matrix defined & explained in the simplest way possible. Besides explaining types of
Matrix theory, EduRev gives you an ample number of questions to practice Matrix tests, examples and also practice UPSC
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Additional Information about Matrix for UPSC Preparation
Matrix Free PDF Download
The Matrix is an invaluable resource that delves deep into the core of the UPSC exam.
These study notes are curated by experts and cover all the essential topics and concepts, making your preparation more efficient and effective.
With the help of these notes, you can grasp complex subjects quickly, revise important points easily,
and reinforce your understanding of key concepts. The study notes are presented in a concise and easy-to-understand manner,
allowing you to optimize your learning process. Whether you're looking for best-recommended books, sample papers, study material,
or toppers' notes, this PDF has got you covered. Download the Matrix now and kickstart your journey towards success in the UPSC exam.
Importance of Matrix
The importance of Matrix cannot be overstated, especially for UPSC aspirants.
This document holds the key to success in the UPSC exam.
It offers a detailed understanding of the concept, providing invaluable insights into the topic.
By knowing the concepts well in advance, students can plan their preparation effectively.
Utilize this indispensable guide for a well-rounded preparation and achieve your desired results.
Matrix Notes
Matrix Notes offer in-depth insights into the specific topic to help you master it with ease.
This comprehensive document covers all aspects related to Matrix.
It includes detailed information about the exam syllabus, recommended books, and study materials for a well-rounded preparation.
Practice papers and question papers enable you to assess your progress effectively.
Additionally, the paper analysis provides valuable tips for tackling the exam strategically.
Access to Toppers' notes gives you an edge in understanding complex concepts.
Whether you're a beginner or aiming for advanced proficiency, Matrix Notes on EduRev are your ultimate resource for success.
Matrix UPSC Questions
The "Matrix UPSC Questions" guide is a valuable resource for all aspiring students preparing for the
UPSC exam. It focuses on providing a wide range of practice questions to help students gauge
their understanding of the exam topics. These questions cover the entire syllabus, ensuring comprehensive preparation.
The guide includes previous years' question papers for students to familiarize themselves with the exam's format and difficulty level.
Additionally, it offers subject-specific question banks, allowing students to focus on weak areas and improve their performance.
Study Matrix on the App
Students of UPSC can study Matrix alongwith tests & analysis from the EduRev app,
which will help them while preparing for their exam. Apart from the Matrix,
students can also utilize the EduRev App for other study materials such as previous year question papers, syllabus, important questions, etc.
The EduRev App will make your learning easier as you can access it from anywhere you want.
The content of Matrix is prepared as per the latest UPSC syllabus.
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