Matrices and Determinants: JEE Mains Previous Year Questions (2021-2024) | Mathematics (Maths) for JEE Main & Advanced PDF Download

 Page 1


 
2024 
Q1 - 2024 (01 Feb Shift 1) 
If A=[
v2 1
-1 v2
],B=[
1 0
1 1
],C=ABA
T
 and X=A
T
C
2
 A, then det?X is equal to : 
(1) 243 
(2) 729 
(3) 27 
(4) 891 
Q2 - 2024 (01 Feb Shift 2) 
Let ?? =?? 2
-?? ?? ?? , where ?? is real matrix of order 2×1 such that the relation ?? ?? ?? =?? 1
 
holds. If ?? is a real number such that the relation AX=?? X holds for some non-zero real 
matrix X of order 2×1, then the sum of squares of all possible values of ?? is equal to : 
Q3 - 2024 (27 Jan Shift 1) 
Consider the matrix ?? (?? )=[
cos??? -sin??? 0
sin??? cos??? 0
0 0 1
]. 
Given below are two statements : 
Statement I: ?? (-?? ) is the inverse of the matrix ?? (?? ). 
Statement II: ?? (?? )?? (?? )=?? (?? +?? ). 
In the light of the above statements, choose the correct answer from the options given 
below 
(1) Statement I is false but Statement II is true 
(2) Both Statement I and Statement II are false 
(3) Statement I is true but Statement II is false 
(4) Both Statement I and Statement II are true 
Page 2


 
2024 
Q1 - 2024 (01 Feb Shift 1) 
If A=[
v2 1
-1 v2
],B=[
1 0
1 1
],C=ABA
T
 and X=A
T
C
2
 A, then det?X is equal to : 
(1) 243 
(2) 729 
(3) 27 
(4) 891 
Q2 - 2024 (01 Feb Shift 2) 
Let ?? =?? 2
-?? ?? ?? , where ?? is real matrix of order 2×1 such that the relation ?? ?? ?? =?? 1
 
holds. If ?? is a real number such that the relation AX=?? X holds for some non-zero real 
matrix X of order 2×1, then the sum of squares of all possible values of ?? is equal to : 
Q3 - 2024 (27 Jan Shift 1) 
Consider the matrix ?? (?? )=[
cos??? -sin??? 0
sin??? cos??? 0
0 0 1
]. 
Given below are two statements : 
Statement I: ?? (-?? ) is the inverse of the matrix ?? (?? ). 
Statement II: ?? (?? )?? (?? )=?? (?? +?? ). 
In the light of the above statements, choose the correct answer from the options given 
below 
(1) Statement I is false but Statement II is true 
(2) Both Statement I and Statement II are false 
(3) Statement I is true but Statement II is false 
(4) Both Statement I and Statement II are true 
 
 
Q4 - 2024 (27 Jan Shift 2) 
Let A be a 2×2 real matrix and ?? be the identity matrix of order 2 . If the roots of the 
equation |?? -???? |=0 be -1 and 3 , then the sum of the diagonal elements of the matrix 
?? 2
 is. 
Q5 - 2024 (29 Jan Shift 1) 
Let ?? =[
1 0 0
0 ?? ?? 0 ?? ?? ] and |2?? |
3
=2
21
 where ?? ,?? ??? , Then a value of ?? is 
(1) 3 
(2) 5 
(3) 17 
(4) 9 
Q6 - 2024 (29 Jan Shift 1) 
Let ?? be a square matrix such that ?? ?? ?? =?? . Then 
1
2
 A[( A+A
T
)
2
+(A-A
T
)
2
] is equal 
to 
(1) A
2
+I 
(2) ?? 3
+?? 
(3) ?? 2
+?? ?? 
(4) ?? 3
+?? ?? 
Q7 - 2024 (29 Jan Shift 2) 
Let ?? =[
2 1 2
6 2 11
3 3 2
] and ?? =[
1 2 0
5 0 2
7 1 5
]. The sum of the prime factors of |P
-1
AP-2I| is 
equal to 
(1) 26 
(2) 27 
(3) 66 
(4) 23 
Page 3


 
2024 
Q1 - 2024 (01 Feb Shift 1) 
If A=[
v2 1
-1 v2
],B=[
1 0
1 1
],C=ABA
T
 and X=A
T
C
2
 A, then det?X is equal to : 
(1) 243 
(2) 729 
(3) 27 
(4) 891 
Q2 - 2024 (01 Feb Shift 2) 
Let ?? =?? 2
-?? ?? ?? , where ?? is real matrix of order 2×1 such that the relation ?? ?? ?? =?? 1
 
holds. If ?? is a real number such that the relation AX=?? X holds for some non-zero real 
matrix X of order 2×1, then the sum of squares of all possible values of ?? is equal to : 
Q3 - 2024 (27 Jan Shift 1) 
Consider the matrix ?? (?? )=[
cos??? -sin??? 0
sin??? cos??? 0
0 0 1
]. 
Given below are two statements : 
Statement I: ?? (-?? ) is the inverse of the matrix ?? (?? ). 
Statement II: ?? (?? )?? (?? )=?? (?? +?? ). 
In the light of the above statements, choose the correct answer from the options given 
below 
(1) Statement I is false but Statement II is true 
(2) Both Statement I and Statement II are false 
(3) Statement I is true but Statement II is false 
(4) Both Statement I and Statement II are true 
 
 
Q4 - 2024 (27 Jan Shift 2) 
Let A be a 2×2 real matrix and ?? be the identity matrix of order 2 . If the roots of the 
equation |?? -???? |=0 be -1 and 3 , then the sum of the diagonal elements of the matrix 
?? 2
 is. 
Q5 - 2024 (29 Jan Shift 1) 
Let ?? =[
1 0 0
0 ?? ?? 0 ?? ?? ] and |2?? |
3
=2
21
 where ?? ,?? ??? , Then a value of ?? is 
(1) 3 
(2) 5 
(3) 17 
(4) 9 
Q6 - 2024 (29 Jan Shift 1) 
Let ?? be a square matrix such that ?? ?? ?? =?? . Then 
1
2
 A[( A+A
T
)
2
+(A-A
T
)
2
] is equal 
to 
(1) A
2
+I 
(2) ?? 3
+?? 
(3) ?? 2
+?? ?? 
(4) ?? 3
+?? ?? 
Q7 - 2024 (29 Jan Shift 2) 
Let ?? =[
2 1 2
6 2 11
3 3 2
] and ?? =[
1 2 0
5 0 2
7 1 5
]. The sum of the prime factors of |P
-1
AP-2I| is 
equal to 
(1) 26 
(2) 27 
(3) 66 
(4) 23 
Q8 - 2024 (30 Jan Shift 2) 
Let ?? =(
?? 0 0
0 ?? 0
0 0 ?? ) be a non-zero 3×3 matrix, where sin??? =?? sin?(?? +
2?? 3
)=
?? sin?(?? +
4?? 3
) ?0,?? ?(0,2?? ). For a square matrix ?? , let trace (M) denote the sum of all 
the diagonal entries of M. Then, 
among the statements: 
(I) Trace (R)=0 
(II) If trace (adj?(adj?(?? ))=0, then ?? has exactly one non-zero entry. 
(1) Both (I) and (II) are true 
(2) Neither (I) nor (II) is true 
(3) Only (II) is true 
(4) Only (I) is true 
Q9 - 2024 (31 Jan Shift 2) 
Let ?? be a 3×3 matrix and det?(?? )=2. If 
n=det?(adj?(adj?(……(adj??? )
?            
2024- times 
))) 
Then the remainder when n is divided by 9 is equal to 
 
Answer Key 
Q1 (2)  ???? (?? ) Q3 (4)  Q4 (10) 
 
Q9 (7) 
Q5 (2)  Q6 (4) Q7 (1)  Q8 (2) 
 
 
 
Page 4


 
2024 
Q1 - 2024 (01 Feb Shift 1) 
If A=[
v2 1
-1 v2
],B=[
1 0
1 1
],C=ABA
T
 and X=A
T
C
2
 A, then det?X is equal to : 
(1) 243 
(2) 729 
(3) 27 
(4) 891 
Q2 - 2024 (01 Feb Shift 2) 
Let ?? =?? 2
-?? ?? ?? , where ?? is real matrix of order 2×1 such that the relation ?? ?? ?? =?? 1
 
holds. If ?? is a real number such that the relation AX=?? X holds for some non-zero real 
matrix X of order 2×1, then the sum of squares of all possible values of ?? is equal to : 
Q3 - 2024 (27 Jan Shift 1) 
Consider the matrix ?? (?? )=[
cos??? -sin??? 0
sin??? cos??? 0
0 0 1
]. 
Given below are two statements : 
Statement I: ?? (-?? ) is the inverse of the matrix ?? (?? ). 
Statement II: ?? (?? )?? (?? )=?? (?? +?? ). 
In the light of the above statements, choose the correct answer from the options given 
below 
(1) Statement I is false but Statement II is true 
(2) Both Statement I and Statement II are false 
(3) Statement I is true but Statement II is false 
(4) Both Statement I and Statement II are true 
 
 
Q4 - 2024 (27 Jan Shift 2) 
Let A be a 2×2 real matrix and ?? be the identity matrix of order 2 . If the roots of the 
equation |?? -???? |=0 be -1 and 3 , then the sum of the diagonal elements of the matrix 
?? 2
 is. 
Q5 - 2024 (29 Jan Shift 1) 
Let ?? =[
1 0 0
0 ?? ?? 0 ?? ?? ] and |2?? |
3
=2
21
 where ?? ,?? ??? , Then a value of ?? is 
(1) 3 
(2) 5 
(3) 17 
(4) 9 
Q6 - 2024 (29 Jan Shift 1) 
Let ?? be a square matrix such that ?? ?? ?? =?? . Then 
1
2
 A[( A+A
T
)
2
+(A-A
T
)
2
] is equal 
to 
(1) A
2
+I 
(2) ?? 3
+?? 
(3) ?? 2
+?? ?? 
(4) ?? 3
+?? ?? 
Q7 - 2024 (29 Jan Shift 2) 
Let ?? =[
2 1 2
6 2 11
3 3 2
] and ?? =[
1 2 0
5 0 2
7 1 5
]. The sum of the prime factors of |P
-1
AP-2I| is 
equal to 
(1) 26 
(2) 27 
(3) 66 
(4) 23 
Q8 - 2024 (30 Jan Shift 2) 
Let ?? =(
?? 0 0
0 ?? 0
0 0 ?? ) be a non-zero 3×3 matrix, where sin??? =?? sin?(?? +
2?? 3
)=
?? sin?(?? +
4?? 3
) ?0,?? ?(0,2?? ). For a square matrix ?? , let trace (M) denote the sum of all 
the diagonal entries of M. Then, 
among the statements: 
(I) Trace (R)=0 
(II) If trace (adj?(adj?(?? ))=0, then ?? has exactly one non-zero entry. 
(1) Both (I) and (II) are true 
(2) Neither (I) nor (II) is true 
(3) Only (II) is true 
(4) Only (I) is true 
Q9 - 2024 (31 Jan Shift 2) 
Let ?? be a 3×3 matrix and det?(?? )=2. If 
n=det?(adj?(adj?(……(adj??? )
?            
2024- times 
))) 
Then the remainder when n is divided by 9 is equal to 
 
Answer Key 
Q1 (2)  ???? (?? ) Q3 (4)  Q4 (10) 
 
Q9 (7) 
Q5 (2)  Q6 (4) Q7 (1)  Q8 (2) 
 
 
 
Solutions 
Q1 
?? =[
v2 1
-1 v2
]?det?(?? )=3 
?? =[
1 0
1 1
]?det?(?? )=1 
Now C=ABA
T
?det?(C)=(dct?(A))
2
xdet?(B) 
|?? |=9 
Now |X|=|A
T
C
2
 A| 
=|A
T
||C|
2
| A| 
=|A|
2
|C|
2
 
=9×81 
=729 
Q2 
A=I
2
-2MM
T
 
A
2
=(I
2
-2MM
T
)(I
2
-2MM
T
) 
=I
2
-2MM
T
-2MM
T
+4MM
T
MM
T
 
=I
2
-4MM
T
+4MM
T
 
=I
2
 
AX=?? X 
A
2
X=?? AX 
X=?? (?? X) 
X=?? 2
X 
X(?? 2
-1)=0 
?? 2
=1 
?? =±1 
Sum of square of all possible values =2 
Page 5


 
2024 
Q1 - 2024 (01 Feb Shift 1) 
If A=[
v2 1
-1 v2
],B=[
1 0
1 1
],C=ABA
T
 and X=A
T
C
2
 A, then det?X is equal to : 
(1) 243 
(2) 729 
(3) 27 
(4) 891 
Q2 - 2024 (01 Feb Shift 2) 
Let ?? =?? 2
-?? ?? ?? , where ?? is real matrix of order 2×1 such that the relation ?? ?? ?? =?? 1
 
holds. If ?? is a real number such that the relation AX=?? X holds for some non-zero real 
matrix X of order 2×1, then the sum of squares of all possible values of ?? is equal to : 
Q3 - 2024 (27 Jan Shift 1) 
Consider the matrix ?? (?? )=[
cos??? -sin??? 0
sin??? cos??? 0
0 0 1
]. 
Given below are two statements : 
Statement I: ?? (-?? ) is the inverse of the matrix ?? (?? ). 
Statement II: ?? (?? )?? (?? )=?? (?? +?? ). 
In the light of the above statements, choose the correct answer from the options given 
below 
(1) Statement I is false but Statement II is true 
(2) Both Statement I and Statement II are false 
(3) Statement I is true but Statement II is false 
(4) Both Statement I and Statement II are true 
 
 
Q4 - 2024 (27 Jan Shift 2) 
Let A be a 2×2 real matrix and ?? be the identity matrix of order 2 . If the roots of the 
equation |?? -???? |=0 be -1 and 3 , then the sum of the diagonal elements of the matrix 
?? 2
 is. 
Q5 - 2024 (29 Jan Shift 1) 
Let ?? =[
1 0 0
0 ?? ?? 0 ?? ?? ] and |2?? |
3
=2
21
 where ?? ,?? ??? , Then a value of ?? is 
(1) 3 
(2) 5 
(3) 17 
(4) 9 
Q6 - 2024 (29 Jan Shift 1) 
Let ?? be a square matrix such that ?? ?? ?? =?? . Then 
1
2
 A[( A+A
T
)
2
+(A-A
T
)
2
] is equal 
to 
(1) A
2
+I 
(2) ?? 3
+?? 
(3) ?? 2
+?? ?? 
(4) ?? 3
+?? ?? 
Q7 - 2024 (29 Jan Shift 2) 
Let ?? =[
2 1 2
6 2 11
3 3 2
] and ?? =[
1 2 0
5 0 2
7 1 5
]. The sum of the prime factors of |P
-1
AP-2I| is 
equal to 
(1) 26 
(2) 27 
(3) 66 
(4) 23 
Q8 - 2024 (30 Jan Shift 2) 
Let ?? =(
?? 0 0
0 ?? 0
0 0 ?? ) be a non-zero 3×3 matrix, where sin??? =?? sin?(?? +
2?? 3
)=
?? sin?(?? +
4?? 3
) ?0,?? ?(0,2?? ). For a square matrix ?? , let trace (M) denote the sum of all 
the diagonal entries of M. Then, 
among the statements: 
(I) Trace (R)=0 
(II) If trace (adj?(adj?(?? ))=0, then ?? has exactly one non-zero entry. 
(1) Both (I) and (II) are true 
(2) Neither (I) nor (II) is true 
(3) Only (II) is true 
(4) Only (I) is true 
Q9 - 2024 (31 Jan Shift 2) 
Let ?? be a 3×3 matrix and det?(?? )=2. If 
n=det?(adj?(adj?(……(adj??? )
?            
2024- times 
))) 
Then the remainder when n is divided by 9 is equal to 
 
Answer Key 
Q1 (2)  ???? (?? ) Q3 (4)  Q4 (10) 
 
Q9 (7) 
Q5 (2)  Q6 (4) Q7 (1)  Q8 (2) 
 
 
 
Solutions 
Q1 
?? =[
v2 1
-1 v2
]?det?(?? )=3 
?? =[
1 0
1 1
]?det?(?? )=1 
Now C=ABA
T
?det?(C)=(dct?(A))
2
xdet?(B) 
|?? |=9 
Now |X|=|A
T
C
2
 A| 
=|A
T
||C|
2
| A| 
=|A|
2
|C|
2
 
=9×81 
=729 
Q2 
A=I
2
-2MM
T
 
A
2
=(I
2
-2MM
T
)(I
2
-2MM
T
) 
=I
2
-2MM
T
-2MM
T
+4MM
T
MM
T
 
=I
2
-4MM
T
+4MM
T
 
=I
2
 
AX=?? X 
A
2
X=?? AX 
X=?? (?? X) 
X=?? 2
X 
X(?? 2
-1)=0 
?? 2
=1 
?? =±1 
Sum of square of all possible values =2 
Q3 
?? (-?? )=[
cos??? sin??? 0
-sin??? cos??? 0
0 0 1
] 
?? (?? )·?? (-?? )=[
1 0 0
0 1 0
0 0 1
]=?? 
Hence statement- I is correct 
Now, checking statement II 
?? (?? )=[
cos??? -sin??? 0
sin??? cos??? 0
0 0 1
] 
?? (?? )·?? (?? )=[
cos?(?? +?? ) -sin?(?? +?? ) 0
sin?(?? +?? ) cos?(?? +?? ) 0
0 0 1
] 
?f(x)·f(y)=f(x+y) 
Hence statement-II is also correct. 
Q4 
|?? -???? |=0 
Roots are -1 and 3 
Sum of roots =tr?(A)=2 
Product of roots =|A|=-3 
Let ?? =[
?? ?? ?? ?? ] 
We have a+d=2 
ad-bc=-3 
?? 2
=[
?? ?? ?? ?? ]×[
?? ?? ?? ?? ]=[
?? 2
+???? ???? +????
???? +???? ???? +?? 2
] 
We need a
2
+bc+bc+d
2
 
=?? 2
+2???? +?? 2
 
=(?? +?? )
2
-2???? +2???? 
=4-2(???? -???? )