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JEE Advanced 2025 Paper 1
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© examsnet.com
Question : 4
Total: 48
Consider the matrix
P
=
(
2
0
0
0
2
0
0
0
3
)
Let the transpose of a matrix
X
be denoted by
X
T
. Then the number of
3
×
3
invertible matrices
Q
with integer entries, such that
Q
−
1
=
Q
T
and
P
Q
=
Q
P
,
[JEE Adv 2025 P1]
32
8
16
24
Validate
Solution:
‌
‌
As
‌
P
Q
=
Q
P
∴
[
2
0
0
0
2
0
0
0
3
]
[
a
b
c
d
e
f
g
h
i
]
=
[
a
b
c
d
e
f
g
h
i
]
[
2
0
0
0
2
0
0
0
3
]
⇒
[
2
a
2
b
2
c
2
d
2
e
2
f
3
g
3
h
3
i
]
=
[
2
a
2
b
3
c
2
d
2
e
3
f
2
g
2
h
3
i
]
‌
⇒
c
=
0
,
f
=
0
,
h
=
0
,
g
=
0
‌
‌
We get
‌
Q
=
[
a
b
0
c
d
0
0
0
e
]
Given
Q
Q
T
=
I
‌
⇒
[
a
b
0
c
d
0
0
0
i
]
[
a
c
0
b
d
0
0
0
i
]
=
[
1
0
0
0
1
0
0
0
1
]
‌
⇒
[
a
2
+
b
2
a
c
+
b
d
0
a
c
+
b
d
c
2
+
d
2
0
0
0
i
2
]
=
[
1
0
0
0
1
0
0
0
1
]
‌
⇒
a
2
+
b
2
=
1
,
a
c
+
b
d
=
0
,
c
2
+
d
2
=
1
,
i
2
=
1
Case I:
(
a
,
b
)
=
(
0
,
1
)
or
(
0
,
−
1
)
,
(
c
,
d
)
=
(
1
,
0
)
or
(
−
1
,
0
)
Case II:
(
a
,
b
)
=
(
1
,
0
)
or
(
−
1
,
0
)
,
(
c
,
d
)
=
(
0
,
1
)
or
(
0
,
−
1
)
Therefore total possible
Q
=
16
© examsnet.com
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