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JEE Main Previous Year Questions (2025): Matrices
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Page 1
JEE Main Previous Year Questions
(2025): Matrices
Q1: Let ?? be a square matrix of order 3 such that ?????? (?? )=-?? and
?????? (???????? (-???????? (?? ?? )))=?? ?? +?? ·?? ????
,?? >?? . Then ?? ?? +?? ?? is equal to ____ .
JEE Main 2025 (Online) 22nd January Morning Shift
Ans: 34
Solution:
As ?? adj ?? =|?? |?? ,det(???? )=?? ?? det??
det(3adj(-6adj(3?? )))=3
3
det(adj(-6adj(3?? )))
=3
3
(-6adj(3?? ))
2
=3
3
(-6)
6
|3?? |
4
=3
9
2
6
·3
12
·(-2)
4
=3
21
·2
10
Now comparing with given condition
2
?? +?? 3
????
=2
10
·3
21
?? +?? =10,???? =21
??? =7,?? =3(?? >?? )
?4?? +2?? =28+6=34
Q2: Let A be a ?? ×?? matrix such that ?? ?? ???? =?? for all nonzero ?? ×?? matrices ?? =
[
?? ?? ?? ]. If ?? [
?? ?? ?? ]=[
?? ?? -?? ],?? [
?? ?? ?? ]=[
?? ?? -?? ], and ?????? (?????? (?? ( ?? +?? )))=?? ?? ?? ?? ?? ?? ,?? ,?? ,?? ??? ,
then ?? ?? +?? ?? +?? ?? is
JEE Main 2025 (Online) 24th January Morning Shift
Ans: 44
Solution:
?? ?? ???? =0
(?????? )(
?? 1
?? 2
?? 3
?? 1
?? 2
?? 3
?? 1
?? 2
?? 3
)(
?? ?? ?? )=0
(?????? )(
?? 1
?? +?? 2
?? +?? 3
?? ?? 1
?? +?? 2
?? +?? 3
?? ?? 1
?? +?? 2
?? +?? 3
?? )=0
?? (?? 1
?? +?? 2
?? +?? 3
?? )+?? (?? 1
?? +?? 2
?? +?? 3
?? )
+?? (?? 1
?? +?? 2
?? +?? 3
?? )=0
?? 1
=0,?? 2
=0?? 3
=0
Page 2
JEE Main Previous Year Questions
(2025): Matrices
Q1: Let ?? be a square matrix of order 3 such that ?????? (?? )=-?? and
?????? (???????? (-???????? (?? ?? )))=?? ?? +?? ·?? ????
,?? >?? . Then ?? ?? +?? ?? is equal to ____ .
JEE Main 2025 (Online) 22nd January Morning Shift
Ans: 34
Solution:
As ?? adj ?? =|?? |?? ,det(???? )=?? ?? det??
det(3adj(-6adj(3?? )))=3
3
det(adj(-6adj(3?? )))
=3
3
(-6adj(3?? ))
2
=3
3
(-6)
6
|3?? |
4
=3
9
2
6
·3
12
·(-2)
4
=3
21
·2
10
Now comparing with given condition
2
?? +?? 3
????
=2
10
·3
21
?? +?? =10,???? =21
??? =7,?? =3(?? >?? )
?4?? +2?? =28+6=34
Q2: Let A be a ?? ×?? matrix such that ?? ?? ???? =?? for all nonzero ?? ×?? matrices ?? =
[
?? ?? ?? ]. If ?? [
?? ?? ?? ]=[
?? ?? -?? ],?? [
?? ?? ?? ]=[
?? ?? -?? ], and ?????? (?????? (?? ( ?? +?? )))=?? ?? ?? ?? ?? ?? ,?? ,?? ,?? ??? ,
then ?? ?? +?? ?? +?? ?? is
JEE Main 2025 (Online) 24th January Morning Shift
Ans: 44
Solution:
?? ?? ???? =0
(?????? )(
?? 1
?? 2
?? 3
?? 1
?? 2
?? 3
?? 1
?? 2
?? 3
)(
?? ?? ?? )=0
(?????? )(
?? 1
?? +?? 2
?? +?? 3
?? ?? 1
?? +?? 2
?? +?? 3
?? ?? 1
?? +?? 2
?? +?? 3
?? )=0
?? (?? 1
?? +?? 2
?? +?? 3
?? )+?? (?? 1
?? +?? 2
?? +?? 3
?? )
+?? (?? 1
?? +?? 2
?? +?? 3
?? )=0
?? 1
=0,?? 2
=0?? 3
=0
?? 2
+?? 1
=0,?? 3
+?? 1
=0,?? 3
=?? 2
=0
?? = skew symm matrix
?? =(
0 ?? ?? -?? 0 ?? -?? -?? 0
); ?? =(
1
1
1
)=(
1
4
-5
)
??? =(
0 ?? ?? -?? 0 ?? -?? -?? 0
)(
1
1
1
)=(
1
4
-5
)
?? +?? =1
-?? +?? =4
?? +?? =5
(
0 ?? ?? -?? 0 ?? -?? -?? 0
)(
1
2
1
)=(
1
4
-8
)
2?? +?? =0 ?? =-1
-?? +?? =4 ?? =2
-?? -2?? =-8 ?? =3
?? =(
0 -1 2
1 0 3
-2 -3 0
)
2( A+I)=(
2 -2 4
2 2 6
-2 -6 2
)
2( A+I)=120?det|adi(2( A+I))|
=120
2
=2
6
·3
2
·5
2
?? =6,?? =2,?? =2
Q3: Let ?? denote the set of all real matrices of order ?? ×?? and let ?? =
{-?? ,-?? ,-?? ,?? ,?? } . Let
?? ?? ={?? =[?? ????
]??? :?? =?? ?? and ?? ????
??? ,???,??} ,
?? ?? ={?? =[?? ????
]??? :?? =-?? ?? and ?? ????
??? ,???,??} ,
?? ?? ={?? =[?? ????
]??? :?? ????
+?? ????
+?? ????
=?? and ?? ????
??? ,???,??} .
If ?? (?? ?? ??? ?? ??? ?? )=?? ???? ?? , then ?? equls ____ .
JEE Main 2025 (Online) 28th January Morning Shift
Ans: 1613
Solution:
[
?? 11
?? 12
?? 13
?? 21
?? 22
?? 23
?? 31
?? 32
?? 33
]
No. of elements in S
1
:A=A
T
?5
3
×5
3
No. of elements in ?? 2
:?? =-?? ?? ?0
since no zero in S
2
No. of elements in S
3
?
Page 3
JEE Main Previous Year Questions
(2025): Matrices
Q1: Let ?? be a square matrix of order 3 such that ?????? (?? )=-?? and
?????? (???????? (-???????? (?? ?? )))=?? ?? +?? ·?? ????
,?? >?? . Then ?? ?? +?? ?? is equal to ____ .
JEE Main 2025 (Online) 22nd January Morning Shift
Ans: 34
Solution:
As ?? adj ?? =|?? |?? ,det(???? )=?? ?? det??
det(3adj(-6adj(3?? )))=3
3
det(adj(-6adj(3?? )))
=3
3
(-6adj(3?? ))
2
=3
3
(-6)
6
|3?? |
4
=3
9
2
6
·3
12
·(-2)
4
=3
21
·2
10
Now comparing with given condition
2
?? +?? 3
????
=2
10
·3
21
?? +?? =10,???? =21
??? =7,?? =3(?? >?? )
?4?? +2?? =28+6=34
Q2: Let A be a ?? ×?? matrix such that ?? ?? ???? =?? for all nonzero ?? ×?? matrices ?? =
[
?? ?? ?? ]. If ?? [
?? ?? ?? ]=[
?? ?? -?? ],?? [
?? ?? ?? ]=[
?? ?? -?? ], and ?????? (?????? (?? ( ?? +?? )))=?? ?? ?? ?? ?? ?? ,?? ,?? ,?? ??? ,
then ?? ?? +?? ?? +?? ?? is
JEE Main 2025 (Online) 24th January Morning Shift
Ans: 44
Solution:
?? ?? ???? =0
(?????? )(
?? 1
?? 2
?? 3
?? 1
?? 2
?? 3
?? 1
?? 2
?? 3
)(
?? ?? ?? )=0
(?????? )(
?? 1
?? +?? 2
?? +?? 3
?? ?? 1
?? +?? 2
?? +?? 3
?? ?? 1
?? +?? 2
?? +?? 3
?? )=0
?? (?? 1
?? +?? 2
?? +?? 3
?? )+?? (?? 1
?? +?? 2
?? +?? 3
?? )
+?? (?? 1
?? +?? 2
?? +?? 3
?? )=0
?? 1
=0,?? 2
=0?? 3
=0
?? 2
+?? 1
=0,?? 3
+?? 1
=0,?? 3
=?? 2
=0
?? = skew symm matrix
?? =(
0 ?? ?? -?? 0 ?? -?? -?? 0
); ?? =(
1
1
1
)=(
1
4
-5
)
??? =(
0 ?? ?? -?? 0 ?? -?? -?? 0
)(
1
1
1
)=(
1
4
-5
)
?? +?? =1
-?? +?? =4
?? +?? =5
(
0 ?? ?? -?? 0 ?? -?? -?? 0
)(
1
2
1
)=(
1
4
-8
)
2?? +?? =0 ?? =-1
-?? +?? =4 ?? =2
-?? -2?? =-8 ?? =3
?? =(
0 -1 2
1 0 3
-2 -3 0
)
2( A+I)=(
2 -2 4
2 2 6
-2 -6 2
)
2( A+I)=120?det|adi(2( A+I))|
=120
2
=2
6
·3
2
·5
2
?? =6,?? =2,?? =2
Q3: Let ?? denote the set of all real matrices of order ?? ×?? and let ?? =
{-?? ,-?? ,-?? ,?? ,?? } . Let
?? ?? ={?? =[?? ????
]??? :?? =?? ?? and ?? ????
??? ,???,??} ,
?? ?? ={?? =[?? ????
]??? :?? =-?? ?? and ?? ????
??? ,???,??} ,
?? ?? ={?? =[?? ????
]??? :?? ????
+?? ????
+?? ????
=?? and ?? ????
??? ,???,??} .
If ?? (?? ?? ??? ?? ??? ?? )=?? ???? ?? , then ?? equls ____ .
JEE Main 2025 (Online) 28th January Morning Shift
Ans: 1613
Solution:
[
?? 11
?? 12
?? 13
?? 21
?? 22
?? 23
?? 31
?? 32
?? 33
]
No. of elements in S
1
:A=A
T
?5
3
×5
3
No. of elements in ?? 2
:?? =-?? ?? ?0
since no zero in S
2
No. of elements in S
3
?
?? 11
+?? 22
+?? 33
=0?(1,2,-3)?31
or
(1,1,-2)?3
or
(-1,-1,2)?3 }
?12×5
6
n(S
1
n S
3
)=12×5
3
n(S
1
? S
2
? S
3
)=5
6
(1+12)-12×5
3
?5
3
×[13×5
3
-12]=125?? ?? =1613
Q4: Let ?? ={?? ??? :?? ?? ?? +?? ?? =?? ?? -?? -?? }, where ?? =[
?? -?? ?? ?? ]. Then ?? (?? ) is equal
to ____ .
JEE Main 2025 (Online) 29th January Morning Shift
Ans: 2
Solution:
?? =[
2 -1
1 0
]
?? 2
=[
3 -2
2 -1
],?? 3
=[
4 -3
3 -2
],?? 4
=[
5 -4
4 -3
]
and so on
A
6
=[
7 -6
6 -5
]
?? ?? =[
?? +1 -?? ?? -?? +1
],
A
m
2
=[
m
2
+1 -m
2
m
2
-(m
2
-1)
]
A
m
2
+A
m
=3I-A
-6
[
m
2
+1 -m
2
m
2
-(m
2
-1)
]+[
m+1 -m
m -m+1
]
=3[
1 0
0 1
]-[
-5 6
-6 7
]
=[
8 -6
6 -4
]
=m
2
+1+m+1=8
=m
2
+m-6=0? m=-3,2
n(s)=2
Q5: Let ?? be the identity matrix of order ?? ×?? and for the matrix ?? =
[
?? ?? ?? ?? ?? ?? ?? -?? ?? ],|?? |=-?? . Let ?? be the inverse of the matrix ?????? (???????? (?? ?? )). Then
|(?? ?? +?? )| is equal to ____
JEE Main 2025 (Online) 3rd April Evening Shift
Page 4
JEE Main Previous Year Questions
(2025): Matrices
Q1: Let ?? be a square matrix of order 3 such that ?????? (?? )=-?? and
?????? (???????? (-???????? (?? ?? )))=?? ?? +?? ·?? ????
,?? >?? . Then ?? ?? +?? ?? is equal to ____ .
JEE Main 2025 (Online) 22nd January Morning Shift
Ans: 34
Solution:
As ?? adj ?? =|?? |?? ,det(???? )=?? ?? det??
det(3adj(-6adj(3?? )))=3
3
det(adj(-6adj(3?? )))
=3
3
(-6adj(3?? ))
2
=3
3
(-6)
6
|3?? |
4
=3
9
2
6
·3
12
·(-2)
4
=3
21
·2
10
Now comparing with given condition
2
?? +?? 3
????
=2
10
·3
21
?? +?? =10,???? =21
??? =7,?? =3(?? >?? )
?4?? +2?? =28+6=34
Q2: Let A be a ?? ×?? matrix such that ?? ?? ???? =?? for all nonzero ?? ×?? matrices ?? =
[
?? ?? ?? ]. If ?? [
?? ?? ?? ]=[
?? ?? -?? ],?? [
?? ?? ?? ]=[
?? ?? -?? ], and ?????? (?????? (?? ( ?? +?? )))=?? ?? ?? ?? ?? ?? ,?? ,?? ,?? ??? ,
then ?? ?? +?? ?? +?? ?? is
JEE Main 2025 (Online) 24th January Morning Shift
Ans: 44
Solution:
?? ?? ???? =0
(?????? )(
?? 1
?? 2
?? 3
?? 1
?? 2
?? 3
?? 1
?? 2
?? 3
)(
?? ?? ?? )=0
(?????? )(
?? 1
?? +?? 2
?? +?? 3
?? ?? 1
?? +?? 2
?? +?? 3
?? ?? 1
?? +?? 2
?? +?? 3
?? )=0
?? (?? 1
?? +?? 2
?? +?? 3
?? )+?? (?? 1
?? +?? 2
?? +?? 3
?? )
+?? (?? 1
?? +?? 2
?? +?? 3
?? )=0
?? 1
=0,?? 2
=0?? 3
=0
?? 2
+?? 1
=0,?? 3
+?? 1
=0,?? 3
=?? 2
=0
?? = skew symm matrix
?? =(
0 ?? ?? -?? 0 ?? -?? -?? 0
); ?? =(
1
1
1
)=(
1
4
-5
)
??? =(
0 ?? ?? -?? 0 ?? -?? -?? 0
)(
1
1
1
)=(
1
4
-5
)
?? +?? =1
-?? +?? =4
?? +?? =5
(
0 ?? ?? -?? 0 ?? -?? -?? 0
)(
1
2
1
)=(
1
4
-8
)
2?? +?? =0 ?? =-1
-?? +?? =4 ?? =2
-?? -2?? =-8 ?? =3
?? =(
0 -1 2
1 0 3
-2 -3 0
)
2( A+I)=(
2 -2 4
2 2 6
-2 -6 2
)
2( A+I)=120?det|adi(2( A+I))|
=120
2
=2
6
·3
2
·5
2
?? =6,?? =2,?? =2
Q3: Let ?? denote the set of all real matrices of order ?? ×?? and let ?? =
{-?? ,-?? ,-?? ,?? ,?? } . Let
?? ?? ={?? =[?? ????
]??? :?? =?? ?? and ?? ????
??? ,???,??} ,
?? ?? ={?? =[?? ????
]??? :?? =-?? ?? and ?? ????
??? ,???,??} ,
?? ?? ={?? =[?? ????
]??? :?? ????
+?? ????
+?? ????
=?? and ?? ????
??? ,???,??} .
If ?? (?? ?? ??? ?? ??? ?? )=?? ???? ?? , then ?? equls ____ .
JEE Main 2025 (Online) 28th January Morning Shift
Ans: 1613
Solution:
[
?? 11
?? 12
?? 13
?? 21
?? 22
?? 23
?? 31
?? 32
?? 33
]
No. of elements in S
1
:A=A
T
?5
3
×5
3
No. of elements in ?? 2
:?? =-?? ?? ?0
since no zero in S
2
No. of elements in S
3
?
?? 11
+?? 22
+?? 33
=0?(1,2,-3)?31
or
(1,1,-2)?3
or
(-1,-1,2)?3 }
?12×5
6
n(S
1
n S
3
)=12×5
3
n(S
1
? S
2
? S
3
)=5
6
(1+12)-12×5
3
?5
3
×[13×5
3
-12]=125?? ?? =1613
Q4: Let ?? ={?? ??? :?? ?? ?? +?? ?? =?? ?? -?? -?? }, where ?? =[
?? -?? ?? ?? ]. Then ?? (?? ) is equal
to ____ .
JEE Main 2025 (Online) 29th January Morning Shift
Ans: 2
Solution:
?? =[
2 -1
1 0
]
?? 2
=[
3 -2
2 -1
],?? 3
=[
4 -3
3 -2
],?? 4
=[
5 -4
4 -3
]
and so on
A
6
=[
7 -6
6 -5
]
?? ?? =[
?? +1 -?? ?? -?? +1
],
A
m
2
=[
m
2
+1 -m
2
m
2
-(m
2
-1)
]
A
m
2
+A
m
=3I-A
-6
[
m
2
+1 -m
2
m
2
-(m
2
-1)
]+[
m+1 -m
m -m+1
]
=3[
1 0
0 1
]-[
-5 6
-6 7
]
=[
8 -6
6 -4
]
=m
2
+1+m+1=8
=m
2
+m-6=0? m=-3,2
n(s)=2
Q5: Let ?? be the identity matrix of order ?? ×?? and for the matrix ?? =
[
?? ?? ?? ?? ?? ?? ?? -?? ?? ],|?? |=-?? . Let ?? be the inverse of the matrix ?????? (???????? (?? ?? )). Then
|(?? ?? +?? )| is equal to ____
JEE Main 2025 (Online) 3rd April Evening Shift
Ans: 38
Solution:
?? =[adj(?? adj(?? 2
))]
-1
Adj(?? 2
)=(adj?? )
2
??? adj(?? 2
)=?? adj(?? )·(adj?? )
=?? (|?? |?? -1
)
2
=|?? |
2
(?? -1
)=?? -1
??? =(adj(?? -1
))
-1
=(|(?? -1
)|?? )
-1
=
?? -1
-1
=-?? -1
??? =-?? -1
|?? |=-1=
?? 2 3
4 5 6
7 -1 2
| =-1??? =3
|3?? +?? |=?? -3?? -1
| =
|?? ||?? -3?? -1
|
|?? |
=
|?? -3?? |
|?? |
=
|?? -3?? |
-1
=
|
0 2 3
4 2 6
7 -1 -1
|
-1
=38
?|3?? +?? |=38
Q6: Let ?? =[
?????? ?? ?? -?????? ?? ?? ?? ?? ?????? ?? ?? ?????? ?? ]. If for some ?? ?(?? ,?? ),?? ?? =?? ?? , then the sum of the
diagonal elements of the matrix (?? +??)
?? +(?? -??)
?? -?? ?? is equal to ____ .
JEE Main 2025 (Online) 4th April Morning Shift
Ans: 6
Solution:
Note that ?? is orthogonal:
?? ?? ?? =?? ?? ?? =?? and ?? ?? =?? -1
Given ?? 2
=?? ?? , then:
?? 3
=??
Tr(?? +?? )
3
+(?? -??)
3
-6?? =Tr(2?? 3
+6?? -6?? )
=Tr(2?? 3
)=Tr(2?? )
( Using (?? +??)
3
+(?? -?? )
3
=2?? 3
+6?? and 2?? 3
=2?? )= 6
Q7: The number of singular matrices of order ?? , whose elements are from the set
{?? ,?? ,?? ,?? } , is ____ .
JEE Main 2025 (Online) 7th April Morning Shift
Solution:
Let ?? =[
?? ?? ?? ?? ]
for ?? to be singular matrix
Page 5
JEE Main Previous Year Questions
(2025): Matrices
Q1: Let ?? be a square matrix of order 3 such that ?????? (?? )=-?? and
?????? (???????? (-???????? (?? ?? )))=?? ?? +?? ·?? ????
,?? >?? . Then ?? ?? +?? ?? is equal to ____ .
JEE Main 2025 (Online) 22nd January Morning Shift
Ans: 34
Solution:
As ?? adj ?? =|?? |?? ,det(???? )=?? ?? det??
det(3adj(-6adj(3?? )))=3
3
det(adj(-6adj(3?? )))
=3
3
(-6adj(3?? ))
2
=3
3
(-6)
6
|3?? |
4
=3
9
2
6
·3
12
·(-2)
4
=3
21
·2
10
Now comparing with given condition
2
?? +?? 3
????
=2
10
·3
21
?? +?? =10,???? =21
??? =7,?? =3(?? >?? )
?4?? +2?? =28+6=34
Q2: Let A be a ?? ×?? matrix such that ?? ?? ???? =?? for all nonzero ?? ×?? matrices ?? =
[
?? ?? ?? ]. If ?? [
?? ?? ?? ]=[
?? ?? -?? ],?? [
?? ?? ?? ]=[
?? ?? -?? ], and ?????? (?????? (?? ( ?? +?? )))=?? ?? ?? ?? ?? ?? ,?? ,?? ,?? ??? ,
then ?? ?? +?? ?? +?? ?? is
JEE Main 2025 (Online) 24th January Morning Shift
Ans: 44
Solution:
?? ?? ???? =0
(?????? )(
?? 1
?? 2
?? 3
?? 1
?? 2
?? 3
?? 1
?? 2
?? 3
)(
?? ?? ?? )=0
(?????? )(
?? 1
?? +?? 2
?? +?? 3
?? ?? 1
?? +?? 2
?? +?? 3
?? ?? 1
?? +?? 2
?? +?? 3
?? )=0
?? (?? 1
?? +?? 2
?? +?? 3
?? )+?? (?? 1
?? +?? 2
?? +?? 3
?? )
+?? (?? 1
?? +?? 2
?? +?? 3
?? )=0
?? 1
=0,?? 2
=0?? 3
=0
?? 2
+?? 1
=0,?? 3
+?? 1
=0,?? 3
=?? 2
=0
?? = skew symm matrix
?? =(
0 ?? ?? -?? 0 ?? -?? -?? 0
); ?? =(
1
1
1
)=(
1
4
-5
)
??? =(
0 ?? ?? -?? 0 ?? -?? -?? 0
)(
1
1
1
)=(
1
4
-5
)
?? +?? =1
-?? +?? =4
?? +?? =5
(
0 ?? ?? -?? 0 ?? -?? -?? 0
)(
1
2
1
)=(
1
4
-8
)
2?? +?? =0 ?? =-1
-?? +?? =4 ?? =2
-?? -2?? =-8 ?? =3
?? =(
0 -1 2
1 0 3
-2 -3 0
)
2( A+I)=(
2 -2 4
2 2 6
-2 -6 2
)
2( A+I)=120?det|adi(2( A+I))|
=120
2
=2
6
·3
2
·5
2
?? =6,?? =2,?? =2
Q3: Let ?? denote the set of all real matrices of order ?? ×?? and let ?? =
{-?? ,-?? ,-?? ,?? ,?? } . Let
?? ?? ={?? =[?? ????
]??? :?? =?? ?? and ?? ????
??? ,???,??} ,
?? ?? ={?? =[?? ????
]??? :?? =-?? ?? and ?? ????
??? ,???,??} ,
?? ?? ={?? =[?? ????
]??? :?? ????
+?? ????
+?? ????
=?? and ?? ????
??? ,???,??} .
If ?? (?? ?? ??? ?? ??? ?? )=?? ???? ?? , then ?? equls ____ .
JEE Main 2025 (Online) 28th January Morning Shift
Ans: 1613
Solution:
[
?? 11
?? 12
?? 13
?? 21
?? 22
?? 23
?? 31
?? 32
?? 33
]
No. of elements in S
1
:A=A
T
?5
3
×5
3
No. of elements in ?? 2
:?? =-?? ?? ?0
since no zero in S
2
No. of elements in S
3
?
?? 11
+?? 22
+?? 33
=0?(1,2,-3)?31
or
(1,1,-2)?3
or
(-1,-1,2)?3 }
?12×5
6
n(S
1
n S
3
)=12×5
3
n(S
1
? S
2
? S
3
)=5
6
(1+12)-12×5
3
?5
3
×[13×5
3
-12]=125?? ?? =1613
Q4: Let ?? ={?? ??? :?? ?? ?? +?? ?? =?? ?? -?? -?? }, where ?? =[
?? -?? ?? ?? ]. Then ?? (?? ) is equal
to ____ .
JEE Main 2025 (Online) 29th January Morning Shift
Ans: 2
Solution:
?? =[
2 -1
1 0
]
?? 2
=[
3 -2
2 -1
],?? 3
=[
4 -3
3 -2
],?? 4
=[
5 -4
4 -3
]
and so on
A
6
=[
7 -6
6 -5
]
?? ?? =[
?? +1 -?? ?? -?? +1
],
A
m
2
=[
m
2
+1 -m
2
m
2
-(m
2
-1)
]
A
m
2
+A
m
=3I-A
-6
[
m
2
+1 -m
2
m
2
-(m
2
-1)
]+[
m+1 -m
m -m+1
]
=3[
1 0
0 1
]-[
-5 6
-6 7
]
=[
8 -6
6 -4
]
=m
2
+1+m+1=8
=m
2
+m-6=0? m=-3,2
n(s)=2
Q5: Let ?? be the identity matrix of order ?? ×?? and for the matrix ?? =
[
?? ?? ?? ?? ?? ?? ?? -?? ?? ],|?? |=-?? . Let ?? be the inverse of the matrix ?????? (???????? (?? ?? )). Then
|(?? ?? +?? )| is equal to ____
JEE Main 2025 (Online) 3rd April Evening Shift
Ans: 38
Solution:
?? =[adj(?? adj(?? 2
))]
-1
Adj(?? 2
)=(adj?? )
2
??? adj(?? 2
)=?? adj(?? )·(adj?? )
=?? (|?? |?? -1
)
2
=|?? |
2
(?? -1
)=?? -1
??? =(adj(?? -1
))
-1
=(|(?? -1
)|?? )
-1
=
?? -1
-1
=-?? -1
??? =-?? -1
|?? |=-1=
?? 2 3
4 5 6
7 -1 2
| =-1??? =3
|3?? +?? |=?? -3?? -1
| =
|?? ||?? -3?? -1
|
|?? |
=
|?? -3?? |
|?? |
=
|?? -3?? |
-1
=
|
0 2 3
4 2 6
7 -1 -1
|
-1
=38
?|3?? +?? |=38
Q6: Let ?? =[
?????? ?? ?? -?????? ?? ?? ?? ?? ?????? ?? ?? ?????? ?? ]. If for some ?? ?(?? ,?? ),?? ?? =?? ?? , then the sum of the
diagonal elements of the matrix (?? +??)
?? +(?? -??)
?? -?? ?? is equal to ____ .
JEE Main 2025 (Online) 4th April Morning Shift
Ans: 6
Solution:
Note that ?? is orthogonal:
?? ?? ?? =?? ?? ?? =?? and ?? ?? =?? -1
Given ?? 2
=?? ?? , then:
?? 3
=??
Tr(?? +?? )
3
+(?? -??)
3
-6?? =Tr(2?? 3
+6?? -6?? )
=Tr(2?? 3
)=Tr(2?? )
( Using (?? +??)
3
+(?? -?? )
3
=2?? 3
+6?? and 2?? 3
=2?? )= 6
Q7: The number of singular matrices of order ?? , whose elements are from the set
{?? ,?? ,?? ,?? } , is ____ .
JEE Main 2025 (Online) 7th April Morning Shift
Solution:
Let ?? =[
?? ?? ?? ?? ]
for ?? to be singular matrix
???? =????
Case 1: exactly 1 number is used ?
4
?? 1
ways
Case 2 : exactly 2 numbers is used ?
4
?? 2
ways
Case 3 : exactly 3 numbers used ? none will be singular.
Case 4: exactly 4 numbers is used
????? =???? ?2×9=3×6
?
4
?? 1
×2!=8 matrix .
? Total ways ?4+6×4+8=36 matrices.
Q8: For a ?? ×?? matrix ?? , let trace (?? ) denote the sum of all the diagonal elements of
?? . Let ?? be a ?? ×?? matrix such that |?? |=
?? ?? and trace (?? )=?? . If ?? =?????? (?????? (?? ?? )) ,
then the value of |?? |+?????????? (?? ) equals :
JEE Main 2025 (Online) 22nd January Evening Shift
Options:
A. 56
B. 132
C. 174
D. 280
Ans: D
Solution:
?? =adj(adj(2?? ))=det(2?? )·(2?? )
Since ?? is a 3×3 matrix with
det(?? )=
1
2
,
the determinant of 2?? is computed as
det(2?? )=2
3
det(?? )=8·
1
2
=4.
Thus,
?? =4·(2?? )=8?? .
Now, compute the determinant and the trace of ?? :
Determinant of ?? :
det(?? )=det(8?? )=8
3
det(?? )=512·
1
2
=256.
Trace of ?? :
trace(?? )=trace(8?? )=8·trace(?? )=8·3=24.
Finally, adding these results:
det(?? )+trace(?? )=256+24=280.
Q9: If ?? ,?? , and (?????? (?? -?? )+?????? (?? -?? )) are non-singular matrices of same order, then
the inverse of ?? (?????? (?? -?? )+?????? (?? -?? ))
-?? ?? , is equal to
JEE Main 2025 (Online) 23rd January Morning Shift
Read MoreFAQs on JEE Main Previous Year Questions (2025): Matrices
| 1. What are the basic operations that can be performed on matrices? |  |
Ans. The basic operations that can be performed on matrices include addition, subtraction, and multiplication. Matrix addition and subtraction can only be performed on matrices of the same dimensions, while matrix multiplication can be performed when the number of columns in the first matrix is equal to the number of rows in the second matrix. Additionally, there is also the operation of finding the transpose of a matrix, which involves flipping the matrix over its diagonal.
| 2. How do you determine if two matrices are equal? | |
Ans. Two matrices are considered equal if they have the same dimensions and all corresponding elements are equal. This means that for two matrices A (of size m x n) and B (of size m x n), A = B if and only if aᵢⱼ = bᵢⱼ for all i (1 to m) and j (1 to n).
| 3. What is the determinant of a matrix, and why is it important? | |
Ans. The determinant of a square matrix is a scalar value that is a function of its elements and provides important properties about the matrix. The determinant can be used to determine whether a matrix is invertible (non-singular) or singular (non-invertible). If the determinant is zero, the matrix does not have an inverse. Additionally, the determinant is used in solving systems of linear equations and in applications related to areas and volumes in geometry.
| 4. Can you explain the concept of the inverse of a matrix? | |
Ans. The inverse of a matrix A is another matrix, denoted as A⁻¹, such that when A is multiplied by A⁻¹, the result is the identity matrix I (i.e., AA⁻¹ = I). Not all matrices have inverses; a matrix must be square and its determinant must be non-zero for it to have an inverse. The inverse is useful in solving linear systems and in various applications across mathematics and engineering.
| 5. What is the role of eigenvalues and eigenvectors in matrix theory? | |
Ans. Eigenvalues and eigenvectors are fundamental concepts in matrix theory. An eigenvector of a square matrix A is a non-zero vector v such that when A is applied to v, the result is a scalar multiple of v (i.e., Av = λv, where λ is the eigenvalue). These concepts are crucial in various applications, including stability analysis, quantum mechanics, and principal component analysis, as they provide insights into the properties of linear transformations represented by matrices.
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