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54Q
01
Consider a subspace
W
=
{
(
x
,
y
,
z
,
w
)
∈
R
4
∣
x
+
y
+
z
=
0
,
y
+
z
=
w
}
W=\{(x,y,z,w)\in\mathbb{R}^4\mid x+y+z=0,\;y+z=w\}
W
=
{(
x
,
y
,
z
,
w
)
∈
R
4
∣
x
+
y
+
z
=
0
,
y
+
z
=
w
}
of
R
4
\mathbb{R}^4
R
4
. Which of the following options describes a basis of
W
W
W
?
Single correct
MEDIUM
4 marks
15 March 2026
02
Find
k
1
k_1
k
1
.
Comprehension
MEDIUM
2 marks
15 March 2026
03
Find
k
2
k_2
k
2
.
Comprehension
MEDIUM
2 marks
15 March 2026
04
Consider the vector space
(
W
,
+
,
⋅
)
(W,+,\cdot)
(
W
,
+
,
⋅
)
with the addition and scalar multiplication operations from the correct option of the previous question. What is the dimension of the vector space?
Comprehension
EASY
4 marks
15 March 2026
05
The trace of a square matrix
A
A
A
is defined as the sum of its diagonal entries. Consider the set
W
W
W
whose elements are
3
×
3
3\times 3
3
×
3
matrices with zero trace, i.e.
W
=
{
[
a
11
a
12
a
13
a
21
a
22
a
23
a
31
a
32
a
33
]
∈
M
3
×
3
(
R
)
∣
a
11
+
a
22
+
a
33
=
0
}
.
W=\left\{\begin{bmatrix}a_{11}&a_{12}&a_{13}\\a_{21}&a_{22}&a_{23}\\a_{31}&a_{32}&a_{33}\end{bmatrix}\in M_{3\times3}(\mathbb{R})\mid a_{11}+a_{22}+a_{33}=0\right\}.
W
=
⎩
⎨
⎧
a
11
a
21
a
31
a
12
a
22
a
32
a
13
a
23
a
33
∈
M
3
×
3
(
R
)
∣
a
11
+
a
22
+
a
33
=
0
⎭
⎬
⎫
.
W
W
W
is a vector space with the standard operations of addition and scalar multiplication. Find the dimension of
W
W
W
.
Numerical
MEDIUM
4 marks
26 October 2025
06
Let
A
A
A
be a
5
×
7
5\times7
5
×
7
matrix. If
n
1
n_1
n
1
and
n
2
n_2
n
2
are the minimum and maximum possible values for the rank of
A
A
A
, respectively, then find the value of
n
2
−
n
1
n_2-n_1
n
2
−
n
1
.
Numerical
EASY
4 marks
26 October 2025
07
Consider the matrix
[
2
−
3
1
−
4
3
1
−
4
5
4
k
−
6
8
]
.
\begin{bmatrix}2&-3&1&-4\\3&1&-4&5\\4&k&-6&8\end{bmatrix}.
2
3
4
−
3
1
k
1
−
4
−
6
−
4
5
8
.
Find the value of
k
k
k
such that the rank of the given matrix is
2
2
2
.
Numerical
MEDIUM
4 marks
26 October 2025
08
For
a
=
0
a=0
a
=
0
, find the value of
b
b
b
such that the set
{
v
1
,
v
2
,
v
3
}
\{v_1,v_2,v_3\}
{
v
1
,
v
2
,
v
3
}
is not a basis of
R
3
\mathbb{R}^3
R
3
.
Comprehension
MEDIUM
3 marks
26 October 2025
09
Which of the following is a basis for
V
=
{
A
∈
M
2
×
2
(
R
)
∣
A
T
=
A
}
V=\{A\in M_{2\times2}(\mathbb{R})\mid A^T=A\}
V
=
{
A
∈
M
2
×
2
(
R
)
∣
A
T
=
A
}
, the vector space of
2
×
2
2\times2
2
×
2
real symmetric matrices?
Single correct
MEDIUM
3 marks
13 July 2025
10
Given a vector space
V
V
V
, which of the following statements are equivalent to the statement that a set
B
⊂
V
B\subset V
B
⊂
V
is a basis?
Multiple correct
MEDIUM
3 marks
13 July 2025
11
Find the rank of
A
A
A
.
Comprehension
EASY
1 marks
13 July 2025
12
Find the maximum possible rank of the matrix
A
A
A
.
Comprehension
EASY
2 marks
13 July 2025
13
Find the value of
k
k
k
for which rank of the matrix
A
A
A
is equal to
2
2
2
.
Comprehension
MEDIUM
2 marks
13 July 2025
14
Let
v
1
,
v
2
,
v
3
v_1,v_2,v_3
v
1
,
v
2
,
v
3
be linearly independent vectors in
R
3
\mathbb{R}^3
R
3
. Let
A
∈
M
3
×
3
(
R
)
A\in M_{3\times3}(\mathbb{R})
A
∈
M
3
×
3
(
R
)
be a matrix such that
v
1
−
v
2
v_1-v_2
v
1
−
v
2
,
v
2
−
v
3
v_2-v_3
v
2
−
v
3
,
v
1
+
v
2
v_1+v_2
v
1
+
v
2
are the columns of
A
A
A
. Let
B
∈
M
3
×
3
(
R
)
B\in M_{3\times3}(\mathbb{R})
B
∈
M
3
×
3
(
R
)
be a matrix such that
v
1
+
v
2
+
v
3
v_1+v_2+v_3
v
1
+
v
2
+
v
3
,
2
v
1
+
3
v
2
2v_1+3v_2
2
v
1
+
3
v
2
,
2
v
3
−
v
2
2v_3-v_2
2
v
3
−
v
2
are the columns of
B
B
B
. Which of the following options is correct?
Single correct
MEDIUM
6 marks
23 February 2025
15
The trace of a square matrix
A
A
A
is defined as the sum of its diagonal entries. Consider the set
W
W
W
whose elements are
3
×
3
3\times3
3
×
3
matrices with zero trace, i.e.,
W
=
{
[
a
11
a
12
a
13
a
21
a
22
a
23
a
31
a
32
a
33
]
∈
M
3
×
3
(
R
)
∣
a
11
+
a
22
+
a
33
=
0
}
.
W=\left\{\begin{bmatrix}a_{11}&a_{12}&a_{13}\\a_{21}&a_{22}&a_{23}\\a_{31}&a_{32}&a_{33}\end{bmatrix}\in M_{3\times3}(\mathbb{R})\mid a_{11}+a_{22}+a_{33}=0\right\}.
W
=
⎩
⎨
⎧
a
11
a
21
a
31
a
12
a
22
a
32
a
13
a
23
a
33
∈
M
3
×
3
(
R
)
∣
a
11
+
a
22
+
a
33
=
0
⎭
⎬
⎫
.
W
W
W
is a vector space with the standard operations of addition and scalar multiplication. Find the dimension of
W
W
W
.
Numerical
MEDIUM
4 marks
23 February 2025
16
Let
A
A
A
be a
5
×
7
5\times7
5
×
7
matrix. If
n
1
n_1
n
1
and
n
2
n_2
n
2
are the minimum and maximum possible values for the rank of
A
A
A
, respectively, then find the value of
n
2
−
n
1
n_2-n_1
n
2
−
n
1
.
Numerical
EASY
4 marks
23 February 2025
17
Consider the matrix
[
2
−
3
1
−
4
3
1
−
4
5
4
k
−
6
8
]
.
\begin{bmatrix}2&-3&1&-4\\3&1&-4&5\\4&k&-6&8\end{bmatrix}.
2
3
4
−
3
1
k
1
−
4
−
6
−
4
5
8
.
Find the value of
k
k
k
such that the rank of the given matrix is
2
2
2
.
Numerical
MEDIUM
4 marks
23 February 2025
18
For
a
=
0
a=0
a
=
0
, find the value of
b
b
b
such that the set
{
v
1
,
v
2
,
v
3
}
\{v_1,v_2,v_3\}
{
v
1
,
v
2
,
v
3
}
is not a basis of
R
3
\mathbb{R}^3
R
3
.
Comprehension
MEDIUM
3 marks
23 February 2025
19
Choose the correct set of matrices that forms a basis for
W
1
W_1
W
1
.
Comprehension
MEDIUM
2 marks
23 February 2025
20
Choose all the correct sets of matrices that span
W
2
W_2
W
2
.
Comprehension
MEDIUM
4 marks
23 February 2025
21
What is the dimension of
W
1
∩
W
2
W_1\cap W_2
W
1
∩
W
2
?
Comprehension
EASY
2 marks
23 February 2025
22
Choose all the correct options from the following.
Multiple correct
MEDIUM
3 marks
27 October 2024
23
W
2
=
span
{
(
1
,
−
1
,
1
)
,
(
4
,
1
,
2
)
,
(
2
,
3
,
0
)
}
W_2=\operatorname{span}\{(1,-1,1),(4,1,2),(2,3,0)\}
W
2
=
span
{(
1
,
−
1
,
1
)
,
(
4
,
1
,
2
)
,
(
2
,
3
,
0
)}
.
Comprehension
MEDIUM
1 marks
27 October 2024
24
W
3
W_3
W
3
is the set of all
2
×
2
2\times2
2
×
2
symmetric matrices.
Comprehension
EASY
1 marks
27 October 2024
25
W
1
=
{
A
∈
M
2
×
2
(
R
)
:
A
T
=
−
A
}
W_1=\{A\in M_{2\times2}(\mathbb{R}):A^T=-A\}
W
1
=
{
A
∈
M
2
×
2
(
R
)
:
A
T
=
−
A
}
.
Comprehension
EASY
1 marks
27 October 2024
26
W
2
W_2
W
2
is the set of all
2
×
2
2\times2
2
×
2
matrices such that the sum of entries in each row is zero.
Comprehension
MEDIUM
1 marks
27 October 2024
27
W
3
W_3
W
3
is the set of all
2
×
2
2\times2
2
×
2
matrices such that the sum of the diagonal entries is zero.
Comprehension
MEDIUM
1 marks
27 October 2024
28
Consider the following subsets of
R
4
\mathbb{R}^4
R
4
.
W
=
span
{
(
2
,
−
1
,
0
,
4
)
,
(
−
1
,
1
,
0
,
3
)
,
(
1
,
2
,
0
,
3
)
,
(
2
,
2
,
0
,
10
)
}
W=\operatorname{span}\{(2,-1,0,4),(-1,1,0,3),(1,2,0,3),(2,2,0,10)\}
W
=
span
{(
2
,
−
1
,
0
,
4
)
,
(
−
1
,
1
,
0
,
3
)
,
(
1
,
2
,
0
,
3
)
,
(
2
,
2
,
0
,
10
)}
,
B
1
=
{
(
1
,
0
,
0
,
0
)
,
(
0
,
1
,
0
,
0
)
,
(
0
,
0
,
0
,
1
)
}
B_1=\{(1,0,0,0),(0,1,0,0),(0,0,0,1)\}
B
1
=
{(
1
,
0
,
0
,
0
)
,
(
0
,
1
,
0
,
0
)
,
(
0
,
0
,
0
,
1
)}
, and
B
2
=
{
(
2
,
−
1
,
0
,
4
)
,
(
−
1
,
1
,
0
,
3
)
,
(
1
,
2
,
0
,
3
)
}
B_2=\{(2,-1,0,4),(-1,1,0,3),(1,2,0,3)\}
B
2
=
{(
2
,
−
1
,
0
,
4
)
,
(
−
1
,
1
,
0
,
3
)
,
(
1
,
2
,
0
,
3
)}
. Select the correct option.
Single correct
MEDIUM
3 marks
07 July 2024
29
U
=
{
(
a
,
b
,
c
,
d
,
e
)
:
a
+
b
+
c
+
d
+
e
=
0
e
x
t
a
n
d
a
,
b
,
c
,
d
,
e
∈
R
}
U=\{(a,b,c,d,e):a+b+c+d+e=0 ext{ and } a,b,c,d,e\in\mathbb{R}\}
U
=
{(
a
,
b
,
c
,
d
,
e
)
:
a
+
b
+
c
+
d
+
e
=
0
e
x
t
an
d
a
,
b
,
c
,
d
,
e
∈
R
}
.
Comprehension
EASY
1 marks
07 July 2024
30
V
=
{
[
a
b
0
c
0
a
+
b
]
:
a
,
b
,
c
∈
R
}
V=\left\{\begin{bmatrix}a&b&0\\c&0&a+b\end{bmatrix}:a,b,c\in\mathbb{R}\right\}
V
=
{
[
a
c
b
0
0
a
+
b
]
:
a
,
b
,
c
∈
R
}
.
Comprehension
EASY
1 marks
07 July 2024
31
W
=
span
{
(
1
,
1
,
1
)
,
(
1
,
0
,
−
1
)
,
(
2
,
1
,
0
)
}
W=\operatorname{span}\{(1,1,1),(1,0,-1),(2,1,0)\}
W
=
span
{(
1
,
1
,
1
)
,
(
1
,
0
,
−
1
)
,
(
2
,
1
,
0
)}
.
Comprehension
EASY
1 marks
07 July 2024
32
Consider the vectors
v
1
=
(
1
,
−
1
,
0
)
v_1=(1,-1,0)
v
1
=
(
1
,
−
1
,
0
)
,
v
2
=
(
2
,
3
,
−
1
)
v_2=(2,3,-1)
v
2
=
(
2
,
3
,
−
1
)
, and
v
3
=
(
a
,
b
,
c
)
v_3=(a,b,c)
v
3
=
(
a
,
b
,
c
)
in
R
3
\mathbb{R}^3
R
3
. Choose the correct options from the following.
Multiple correct
MEDIUM
3 marks
25 February 2024
33
W
2
=
{
[
a
0
0
0
b
0
0
0
c
]
:
a
,
b
,
c
∈
R
such that
a
=
b
=
c
}
W_2=\left\{\begin{bmatrix}a&0&0\\0&b&0\\0&0&c\end{bmatrix}:a,b,c\in\mathbb{R}\text{ such that }a=b=c\right\}
W
2
=
⎩
⎨
⎧
a
0
0
0
b
0
0
0
c
:
a
,
b
,
c
∈
R
such that
a
=
b
=
c
⎭
⎬
⎫
. If
W
2
W_2
W
2
is a subspace, find the dimension; else write the answer as
0
0
0
.
Comprehension
EASY
1 marks
25 February 2024
34
W
3
=
{
[
a
0
0
0
b
0
0
0
c
]
:
a
,
b
,
c
∈
R
}
W_3=\left\{\begin{bmatrix}a&0&0\\0&b&0\\0&0&c\end{bmatrix}:a,b,c\in\mathbb{R}\right\}
W
3
=
⎩
⎨
⎧
a
0
0
0
b
0
0
0
c
:
a
,
b
,
c
∈
R
⎭
⎬
⎫
. If
W
3
W_3
W
3
is a subspace, find the dimension; else write the answer as
0
0
0
.
Comprehension
EASY
1 marks
25 February 2024
35
Choose the correct option(s) from the following statements.
Comprehension
MEDIUM
3 marks
29 October 2023
36
Find
dim
(
W
1
+
W
2
)
\dim(W_1+W_2)
dim
(
W
1
+
W
2
)
.
Comprehension
EASY
1 marks
29 October 2023
37
Let
A
∈
W
A\in W
A
∈
W
be a non-zero matrix. Then find
rank
(
A
)
\operatorname{rank}(A)
rank
(
A
)
.
Comprehension
EASY
1 marks
29 October 2023
38
What is the dimension of the vector space
W
W
W
?
Comprehension
EASY
1 marks
29 October 2023
39
Which of the following sets form a basis for
W
W
W
?
Comprehension
EASY
1 marks
29 October 2023
40
What is the rank of
A
A
A
?
Comprehension
EASY
1 marks
29 October 2023
41
V
1
=
{
(
x
,
y
,
z
)
∈
R
3
:
2
x
+
3
y
=
0
=
2
z
+
3
x
}
V_1=\{(x,y,z)\in\mathbb{R}^3:2x+3y=0=2z+3x\}
V
1
=
{(
x
,
y
,
z
)
∈
R
3
:
2
x
+
3
y
=
0
=
2
z
+
3
x
}
with usual addition and scalar multiplication. Find
dim
(
V
1
)
\dim(V_1)
dim
(
V
1
)
.
Comprehension
EASY
1 marks
16 July 2023
42
V
2
=
{
A
∈
M
3
(
R
)
:
sum of the diagonal entries of
A
is
0
and sum of each row is
0
}
V_2=\{A\in M_3(\mathbb{R}):\text{sum of the diagonal entries of }A\text{ is }0\text{ and sum of each row is }0\}
V
2
=
{
A
∈
M
3
(
R
)
:
sum of the diagonal entries of
A
is
0
and sum of each row is
0
}
with usual addition and scalar multiplication of matrices. Find
dim
(
V
2
)
\dim(V_2)
dim
(
V
2
)
.
Comprehension
MEDIUM
2 marks
16 July 2023
43
What is the rank of
A
A
A
?
Comprehension
EASY
1 marks
16 July 2023
44
Subset 1 is a subspace of dimension __________. Enter the numerical value only.
Comprehension
EASY
1 marks
16 October 2022
45
Subset 2 is a subspace of dimension __________. Enter the numerical value only.
Comprehension
EASY
1 marks
16 October 2022
46
Subset 3 is a subspace of dimension __________. Enter the numerical value only.
Comprehension
EASY
1 marks
16 October 2022
47
Subset 4 is a subspace of dimension __________. Enter the numerical value only.
Comprehension
EASY
1 marks
16 October 2022
48
Which of the following option(s) represent
W
1
∩
W
2
W_1\cap W_2
W
1
∩
W
2
?
Comprehension
MEDIUM
2 marks
16 October 2022
49
NOTE: Enter your answer to the nearest integer. What is the dimension of
W
1
∩
W
2
W_1\cap W_2
W
1
∩
W
2
?
Comprehension
EASY
1 marks
16 October 2022
50
If
W
=
Span
{
(
1
,
1
,
−
1
)
,
(
3
,
−
2
,
0
)
,
(
5
,
0
,
−
2
)
,
(
0
,
5
,
−
3
)
}
W=\operatorname{Span}\{(1,1,-1),(3,-2,0),(5,0,-2),(0,5,-3)\}
W
=
Span
{(
1
,
1
,
−
1
)
,
(
3
,
−
2
,
0
)
,
(
5
,
0
,
−
2
)
,
(
0
,
5
,
−
3
)}
, then find the dimension of
W
W
W
.
Numerical
MEDIUM
4 marks
05 June 2022
51
What is the dimension of
W
1
∩
W
2
W_1\cap W_2
W
1
∩
W
2
?
Comprehension
EASY
2 marks
05 June 2022
52
What is the dimension of
W
1
W_1
W
1
?
Comprehension
EASY
2 marks
05 June 2022
53
S
=
{
(
1
,
0
,
1
)
,
(
0
,
1
,
1
)
,
(
1
,
1
,
0
)
}
S=\{(1,0,1),(0,1,1),(1,1,0)\}
S
=
{(
1
,
0
,
1
)
,
(
0
,
1
,
1
)
,
(
1
,
1
,
0
)}
is a ______ of
R
3
\mathbb{R}^3
R
3
. Enter
3
3
3
best possible options as serial numbers in increasing order without commas or spaces.
Comprehension
MEDIUM
2 marks
05 June 2022
54
A spanning set of
R
2
\mathbb{R}^2
R
2
with
2
2
2
elements must be a ______. Enter
2
2
2
best possible options as serial numbers in increasing order without commas or spaces.
Comprehension
EASY
2 marks
05 June 2022
Showing 54 questions.
Mathematics 2 > Week 4
All PYQs
Topic Wise PYQs
Start Weekly Test
Type:
All
Difficulty:
All
Year:
All
54Q
01
Consider a subspace
W
=
{
(
x
,
y
,
z
,
w
)
∈
R
4
∣
x
+
y
+
z
=
0
,
y
+
z
=
w
}
W=\{(x,y,z,w)\in\mathbb{R}^4\mid x+y+z=0,\;y+z=w\}
W
=
{(
x
,
y
,
z
,
w
)
∈
R
4
∣
x
+
y
+
z
=
0
,
y
+
z
=
w
}
of
R
4
\mathbb{R}^4
R
4
. Which of the following options describes a basis of
W
W
W
?
Single correct
MEDIUM
4 marks
15 March 2026
02
Find
k
1
k_1
k
1
.
Comprehension
MEDIUM
2 marks
15 March 2026
03
Find
k
2
k_2
k
2
.
Comprehension
MEDIUM
2 marks
15 March 2026
04
Consider the vector space
(
W
,
+
,
⋅
)
(W,+,\cdot)
(
W
,
+
,
⋅
)
with the addition and scalar multiplication operations from the correct option of the previous question. What is the dimension of the vector space?
Comprehension
EASY
4 marks
15 March 2026
05
The trace of a square matrix
A
A
A
is defined as the sum of its diagonal entries. Consider the set
W
W
W
whose elements are
3
×
3
3\times 3
3
×
3
matrices with zero trace, i.e.
W
=
{
[
a
11
a
12
a
13
a
21
a
22
a
23
a
31
a
32
a
33
]
∈
M
3
×
3
(
R
)
∣
a
11
+
a
22
+
a
33
=
0
}
.
W=\left\{\begin{bmatrix}a_{11}&a_{12}&a_{13}\\a_{21}&a_{22}&a_{23}\\a_{31}&a_{32}&a_{33}\end{bmatrix}\in M_{3\times3}(\mathbb{R})\mid a_{11}+a_{22}+a_{33}=0\right\}.
W
=
⎩
⎨
⎧
a
11
a
21
a
31
a
12
a
22
a
32
a
13
a
23
a
33
∈
M
3
×
3
(
R
)
∣
a
11
+
a
22
+
a
33
=
0
⎭
⎬
⎫
.
W
W
W
is a vector space with the standard operations of addition and scalar multiplication. Find the dimension of
W
W
W
.
Numerical
MEDIUM
4 marks
26 October 2025
06
Let
A
A
A
be a
5
×
7
5\times7
5
×
7
matrix. If
n
1
n_1
n
1
and
n
2
n_2
n
2
are the minimum and maximum possible values for the rank of
A
A
A
, respectively, then find the value of
n
2
−
n
1
n_2-n_1
n
2
−
n
1
.
Numerical
EASY
4 marks
26 October 2025
07
Consider the matrix
[
2
−
3
1
−
4
3
1
−
4
5
4
k
−
6
8
]
.
\begin{bmatrix}2&-3&1&-4\\3&1&-4&5\\4&k&-6&8\end{bmatrix}.
2
3
4
−
3
1
k
1
−
4
−
6
−
4
5
8
.
Find the value of
k
k
k
such that the rank of the given matrix is
2
2
2
.
Numerical
MEDIUM
4 marks
26 October 2025
08
For
a
=
0
a=0
a
=
0
, find the value of
b
b
b
such that the set
{
v
1
,
v
2
,
v
3
}
\{v_1,v_2,v_3\}
{
v
1
,
v
2
,
v
3
}
is not a basis of
R
3
\mathbb{R}^3
R
3
.
Comprehension
MEDIUM
3 marks
26 October 2025
09
Which of the following is a basis for
V
=
{
A
∈
M
2
×
2
(
R
)
∣
A
T
=
A
}
V=\{A\in M_{2\times2}(\mathbb{R})\mid A^T=A\}
V
=
{
A
∈
M
2
×
2
(
R
)
∣
A
T
=
A
}
, the vector space of
2
×
2
2\times2
2
×
2
real symmetric matrices?
Single correct
MEDIUM
3 marks
13 July 2025
10
Given a vector space
V
V
V
, which of the following statements are equivalent to the statement that a set
B
⊂
V
B\subset V
B
⊂
V
is a basis?
Multiple correct
MEDIUM
3 marks
13 July 2025
11
Find the rank of
A
A
A
.
Comprehension
EASY
1 marks
13 July 2025
12
Find the maximum possible rank of the matrix
A
A
A
.
Comprehension
EASY
2 marks
13 July 2025
13
Find the value of
k
k
k
for which rank of the matrix
A
A
A
is equal to
2
2
2
.
Comprehension
MEDIUM
2 marks
13 July 2025
14
Let
v
1
,
v
2
,
v
3
v_1,v_2,v_3
v
1
,
v
2
,
v
3
be linearly independent vectors in
R
3
\mathbb{R}^3
R
3
. Let
A
∈
M
3
×
3
(
R
)
A\in M_{3\times3}(\mathbb{R})
A
∈
M
3
×
3
(
R
)
be a matrix such that
v
1
−
v
2
v_1-v_2
v
1
−
v
2
,
v
2
−
v
3
v_2-v_3
v
2
−
v
3
,
v
1
+
v
2
v_1+v_2
v
1
+
v
2
are the columns of
A
A
A
. Let
B
∈
M
3
×
3
(
R
)
B\in M_{3\times3}(\mathbb{R})
B
∈
M
3
×
3
(
R
)
be a matrix such that
v
1
+
v
2
+
v
3
v_1+v_2+v_3
v
1
+
v
2
+
v
3
,
2
v
1
+
3
v
2
2v_1+3v_2
2
v
1
+
3
v
2
,
2
v
3
−
v
2
2v_3-v_2
2
v
3
−
v
2
are the columns of
B
B
B
. Which of the following options is correct?
Single correct
MEDIUM
6 marks
23 February 2025
15
The trace of a square matrix
A
A
A
is defined as the sum of its diagonal entries. Consider the set
W
W
W
whose elements are
3
×
3
3\times3
3
×
3
matrices with zero trace, i.e.,
W
=
{
[
a
11
a
12
a
13
a
21
a
22
a
23
a
31
a
32
a
33
]
∈
M
3
×
3
(
R
)
∣
a
11
+
a
22
+
a
33
=
0
}
.
W=\left\{\begin{bmatrix}a_{11}&a_{12}&a_{13}\\a_{21}&a_{22}&a_{23}\\a_{31}&a_{32}&a_{33}\end{bmatrix}\in M_{3\times3}(\mathbb{R})\mid a_{11}+a_{22}+a_{33}=0\right\}.
W
=
⎩
⎨
⎧
a
11
a
21
a
31
a
12
a
22
a
32
a
13
a
23
a
33
∈
M
3
×
3
(
R
)
∣
a
11
+
a
22
+
a
33
=
0
⎭
⎬
⎫
.
W
W
W
is a vector space with the standard operations of addition and scalar multiplication. Find the dimension of
W
W
W
.
Numerical
MEDIUM
4 marks
23 February 2025
16
Let
A
A
A
be a
5
×
7
5\times7
5
×
7
matrix. If
n
1
n_1
n
1
and
n
2
n_2
n
2
are the minimum and maximum possible values for the rank of
A
A
A
, respectively, then find the value of
n
2
−
n
1
n_2-n_1
n
2
−
n
1
.
Numerical
EASY
4 marks
23 February 2025
17
Consider the matrix
[
2
−
3
1
−
4
3
1
−
4
5
4
k
−
6
8
]
.
\begin{bmatrix}2&-3&1&-4\\3&1&-4&5\\4&k&-6&8\end{bmatrix}.
2
3
4
−
3
1
k
1
−
4
−
6
−
4
5
8
.
Find the value of
k
k
k
such that the rank of the given matrix is
2
2
2
.
Numerical
MEDIUM
4 marks
23 February 2025
18
For
a
=
0
a=0
a
=
0
, find the value of
b
b
b
such that the set
{
v
1
,
v
2
,
v
3
}
\{v_1,v_2,v_3\}
{
v
1
,
v
2
,
v
3
}
is not a basis of
R
3
\mathbb{R}^3
R
3
.
Comprehension
MEDIUM
3 marks
23 February 2025
19
Choose the correct set of matrices that forms a basis for
W
1
W_1
W
1
.
Comprehension
MEDIUM
2 marks
23 February 2025
20
Choose all the correct sets of matrices that span
W
2
W_2
W
2
.
Comprehension
MEDIUM
4 marks
23 February 2025
21
What is the dimension of
W
1
∩
W
2
W_1\cap W_2
W
1
∩
W
2
?
Comprehension
EASY
2 marks
23 February 2025
22
Choose all the correct options from the following.
Multiple correct
MEDIUM
3 marks
27 October 2024
23
W
2
=
span
{
(
1
,
−
1
,
1
)
,
(
4
,
1
,
2
)
,
(
2
,
3
,
0
)
}
W_2=\operatorname{span}\{(1,-1,1),(4,1,2),(2,3,0)\}
W
2
=
span
{(
1
,
−
1
,
1
)
,
(
4
,
1
,
2
)
,
(
2
,
3
,
0
)}
.
Comprehension
MEDIUM
1 marks
27 October 2024
24
W
3
W_3
W
3
is the set of all
2
×
2
2\times2
2
×
2
symmetric matrices.
Comprehension
EASY
1 marks
27 October 2024
25
W
1
=
{
A
∈
M
2
×
2
(
R
)
:
A
T
=
−
A
}
W_1=\{A\in M_{2\times2}(\mathbb{R}):A^T=-A\}
W
1
=
{
A
∈
M
2
×
2
(
R
)
:
A
T
=
−
A
}
.
Comprehension
EASY
1 marks
27 October 2024
26
W
2
W_2
W
2
is the set of all
2
×
2
2\times2
2
×
2
matrices such that the sum of entries in each row is zero.
Comprehension
MEDIUM
1 marks
27 October 2024
27
W
3
W_3
W
3
is the set of all
2
×
2
2\times2
2
×
2
matrices such that the sum of the diagonal entries is zero.
Comprehension
MEDIUM
1 marks
27 October 2024
28
Consider the following subsets of
R
4
\mathbb{R}^4
R
4
.
W
=
span
{
(
2
,
−
1
,
0
,
4
)
,
(
−
1
,
1
,
0
,
3
)
,
(
1
,
2
,
0
,
3
)
,
(
2
,
2
,
0
,
10
)
}
W=\operatorname{span}\{(2,-1,0,4),(-1,1,0,3),(1,2,0,3),(2,2,0,10)\}
W
=
span
{(
2
,
−
1
,
0
,
4
)
,
(
−
1
,
1
,
0
,
3
)
,
(
1
,
2
,
0
,
3
)
,
(
2
,
2
,
0
,
10
)}
,
B
1
=
{
(
1
,
0
,
0
,
0
)
,
(
0
,
1
,
0
,
0
)
,
(
0
,
0
,
0
,
1
)
}
B_1=\{(1,0,0,0),(0,1,0,0),(0,0,0,1)\}
B
1
=
{(
1
,
0
,
0
,
0
)
,
(
0
,
1
,
0
,
0
)
,
(
0
,
0
,
0
,
1
)}
, and
B
2
=
{
(
2
,
−
1
,
0
,
4
)
,
(
−
1
,
1
,
0
,
3
)
,
(
1
,
2
,
0
,
3
)
}
B_2=\{(2,-1,0,4),(-1,1,0,3),(1,2,0,3)\}
B
2
=
{(
2
,
−
1
,
0
,
4
)
,
(
−
1
,
1
,
0
,
3
)
,
(
1
,
2
,
0
,
3
)}
. Select the correct option.
Single correct
MEDIUM
3 marks
07 July 2024
29
U
=
{
(
a
,
b
,
c
,
d
,
e
)
:
a
+
b
+
c
+
d
+
e
=
0
e
x
t
a
n
d
a
,
b
,
c
,
d
,
e
∈
R
}
U=\{(a,b,c,d,e):a+b+c+d+e=0 ext{ and } a,b,c,d,e\in\mathbb{R}\}
U
=
{(
a
,
b
,
c
,
d
,
e
)
:
a
+
b
+
c
+
d
+
e
=
0
e
x
t
an
d
a
,
b
,
c
,
d
,
e
∈
R
}
.
Comprehension
EASY
1 marks
07 July 2024
30
V
=
{
[
a
b
0
c
0
a
+
b
]
:
a
,
b
,
c
∈
R
}
V=\left\{\begin{bmatrix}a&b&0\\c&0&a+b\end{bmatrix}:a,b,c\in\mathbb{R}\right\}
V
=
{
[
a
c
b
0
0
a
+
b
]
:
a
,
b
,
c
∈
R
}
.
Comprehension
EASY
1 marks
07 July 2024
31
W
=
span
{
(
1
,
1
,
1
)
,
(
1
,
0
,
−
1
)
,
(
2
,
1
,
0
)
}
W=\operatorname{span}\{(1,1,1),(1,0,-1),(2,1,0)\}
W
=
span
{(
1
,
1
,
1
)
,
(
1
,
0
,
−
1
)
,
(
2
,
1
,
0
)}
.
Comprehension
EASY
1 marks
07 July 2024
32
Consider the vectors
v
1
=
(
1
,
−
1
,
0
)
v_1=(1,-1,0)
v
1
=
(
1
,
−
1
,
0
)
,
v
2
=
(
2
,
3
,
−
1
)
v_2=(2,3,-1)
v
2
=
(
2
,
3
,
−
1
)
, and
v
3
=
(
a
,
b
,
c
)
v_3=(a,b,c)
v
3
=
(
a
,
b
,
c
)
in
R
3
\mathbb{R}^3
R
3
. Choose the correct options from the following.
Multiple correct
MEDIUM
3 marks
25 February 2024
33
W
2
=
{
[
a
0
0
0
b
0
0
0
c
]
:
a
,
b
,
c
∈
R
such that
a
=
b
=
c
}
W_2=\left\{\begin{bmatrix}a&0&0\\0&b&0\\0&0&c\end{bmatrix}:a,b,c\in\mathbb{R}\text{ such that }a=b=c\right\}
W
2
=
⎩
⎨
⎧
a
0
0
0
b
0
0
0
c
:
a
,
b
,
c
∈
R
such that
a
=
b
=
c
⎭
⎬
⎫
. If
W
2
W_2
W
2
is a subspace, find the dimension; else write the answer as
0
0
0
.
Comprehension
EASY
1 marks
25 February 2024
34
W
3
=
{
[
a
0
0
0
b
0
0
0
c
]
:
a
,
b
,
c
∈
R
}
W_3=\left\{\begin{bmatrix}a&0&0\\0&b&0\\0&0&c\end{bmatrix}:a,b,c\in\mathbb{R}\right\}
W
3
=
⎩
⎨
⎧
a
0
0
0
b
0
0
0
c
:
a
,
b
,
c
∈
R
⎭
⎬
⎫
. If
W
3
W_3
W
3
is a subspace, find the dimension; else write the answer as
0
0
0
.
Comprehension
EASY
1 marks
25 February 2024
35
Choose the correct option(s) from the following statements.
Comprehension
MEDIUM
3 marks
29 October 2023
36
Find
dim
(
W
1
+
W
2
)
\dim(W_1+W_2)
dim
(
W
1
+
W
2
)
.
Comprehension
EASY
1 marks
29 October 2023
37
Let
A
∈
W
A\in W
A
∈
W
be a non-zero matrix. Then find
rank
(
A
)
\operatorname{rank}(A)
rank
(
A
)
.
Comprehension
EASY
1 marks
29 October 2023
38
What is the dimension of the vector space
W
W
W
?
Comprehension
EASY
1 marks
29 October 2023
39
Which of the following sets form a basis for
W
W
W
?
Comprehension
EASY
1 marks
29 October 2023
40
What is the rank of
A
A
A
?
Comprehension
EASY
1 marks
29 October 2023
41
V
1
=
{
(
x
,
y
,
z
)
∈
R
3
:
2
x
+
3
y
=
0
=
2
z
+
3
x
}
V_1=\{(x,y,z)\in\mathbb{R}^3:2x+3y=0=2z+3x\}
V
1
=
{(
x
,
y
,
z
)
∈
R
3
:
2
x
+
3
y
=
0
=
2
z
+
3
x
}
with usual addition and scalar multiplication. Find
dim
(
V
1
)
\dim(V_1)
dim
(
V
1
)
.
Comprehension
EASY
1 marks
16 July 2023
42
V
2
=
{
A
∈
M
3
(
R
)
:
sum of the diagonal entries of
A
is
0
and sum of each row is
0
}
V_2=\{A\in M_3(\mathbb{R}):\text{sum of the diagonal entries of }A\text{ is }0\text{ and sum of each row is }0\}
V
2
=
{
A
∈
M
3
(
R
)
:
sum of the diagonal entries of
A
is
0
and sum of each row is
0
}
with usual addition and scalar multiplication of matrices. Find
dim
(
V
2
)
\dim(V_2)
dim
(
V
2
)
.
Comprehension
MEDIUM
2 marks
16 July 2023
43
What is the rank of
A
A
A
?
Comprehension
EASY
1 marks
16 July 2023
44
Subset 1 is a subspace of dimension __________. Enter the numerical value only.
Comprehension
EASY
1 marks
16 October 2022
45
Subset 2 is a subspace of dimension __________. Enter the numerical value only.
Comprehension
EASY
1 marks
16 October 2022
46
Subset 3 is a subspace of dimension __________. Enter the numerical value only.
Comprehension
EASY
1 marks
16 October 2022
47
Subset 4 is a subspace of dimension __________. Enter the numerical value only.
Comprehension
EASY
1 marks
16 October 2022
48
Which of the following option(s) represent
W
1
∩
W
2
W_1\cap W_2
W
1
∩
W
2
?
Comprehension
MEDIUM
2 marks
16 October 2022
49
NOTE: Enter your answer to the nearest integer. What is the dimension of
W
1
∩
W
2
W_1\cap W_2
W
1
∩
W
2
?
Comprehension
EASY
1 marks
16 October 2022
50
If
W
=
Span
{
(
1
,
1
,
−
1
)
,
(
3
,
−
2
,
0
)
,
(
5
,
0
,
−
2
)
,
(
0
,
5
,
−
3
)
}
W=\operatorname{Span}\{(1,1,-1),(3,-2,0),(5,0,-2),(0,5,-3)\}
W
=
Span
{(
1
,
1
,
−
1
)
,
(
3
,
−
2
,
0
)
,
(
5
,
0
,
−
2
)
,
(
0
,
5
,
−
3
)}
, then find the dimension of
W
W
W
.
Numerical
MEDIUM
4 marks
05 June 2022
51
What is the dimension of
W
1
∩
W
2
W_1\cap W_2
W
1
∩
W
2
?
Comprehension
EASY
2 marks
05 June 2022
52
What is the dimension of
W
1
W_1
W
1
?
Comprehension
EASY
2 marks
05 June 2022
53
S
=
{
(
1
,
0
,
1
)
,
(
0
,
1
,
1
)
,
(
1
,
1
,
0
)
}
S=\{(1,0,1),(0,1,1),(1,1,0)\}
S
=
{(
1
,
0
,
1
)
,
(
0
,
1
,
1
)
,
(
1
,
1
,
0
)}
is a ______ of
R
3
\mathbb{R}^3
R
3
. Enter
3
3
3
best possible options as serial numbers in increasing order without commas or spaces.
Comprehension
MEDIUM
2 marks
05 June 2022
54
A spanning set of
R
2
\mathbb{R}^2
R
2
with
2
2
2
elements must be a ______. Enter
2
2
2
best possible options as serial numbers in increasing order without commas or spaces.
Comprehension
EASY
2 marks
05 June 2022
Showing 54 questions.
