Matrices and Determinants: JEE Mains Previous Year Questions (2021-2024)

# 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
?
2024- times
)))
Then the remainder when n is divided by 9 is equal to

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
?
2024- times
)))
Then the remainder when n is divided by 9 is equal to

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
?
2024- times
)))
Then the remainder when n is divided by 9 is equal to

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
?? 2
=[
?? ?? ?? ?? ]×[
?? ?? ?? ?? ]=[
?? 2
+???? ???? +????
???? +???? ???? +?? 2
]
We need a
2
+bc+bc+d
2

=?? 2
+2???? +?? 2

=(?? +?? )
2
-2???? +2????
=4-2(???? -???? )
```

## Mathematics (Maths) for JEE Main & Advanced

209 videos|443 docs|143 tests

## FAQs on Matrices and Determinants: JEE Mains Previous Year Questions (2021-2024) - Mathematics (Maths) for JEE Main & Advanced

 1. What are the basic properties of matrices and determinants?
Ans. Matrices and determinants have properties such as commutativity, associativity, distributivity, and the property of the identity element. Matrices also have properties like addition, subtraction, scalar multiplication, and matrix multiplication.
 2. How are matrices and determinants used in solving systems of equations?
Ans. Matrices are used to represent systems of linear equations, and determinants are used to find the solutions to these systems. By performing operations on matrices and determinants, we can solve complex systems of equations efficiently.
 3. Can determinants be used to find the area of a triangle?
Ans. Yes, determinants can be used to find the area of a triangle by representing the three vertices of the triangle as a matrix. By calculating the determinant of this matrix, we can determine the area of the triangle.
 4. How are matrices and determinants applied in computer graphics?
Ans. Matrices are used to represent transformations such as translation, rotation, and scaling in computer graphics. Determinants are used to determine if a transformation will preserve orientation or flip it.
 5. What is the significance of eigenvalues and eigenvectors in matrices?
Ans. Eigenvalues and eigenvectors play a crucial role in matrices as they represent special directions along which a transformation only stretches or compresses, without changing its direction. They are used in various applications such as solving differential equations and analyzing complex systems.

## Mathematics (Maths) for JEE Main & Advanced

209 videos|443 docs|143 tests

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