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JEE Main Math Class 12 Matrices Part 1 Questions
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© examsnet.com
Question : 47
Total: 100
If the matrix
A
=
(
0
2
K
−
1
)
satisfies
A
(
A
3
+
3
I
)
=
2
I
, then the value of K is
[27 Aug 2021 Shift 1]
1
2
−
1
2
−
1
1
Validate
Solution:
Given matrix,
A
=
[
0
2
K
−
1
]
Characteristic equation of A is
|
A
−
λ
I
|
=
0
⇒
|
−
λ
2
K
−
1
−
λ
|
=
0
⇒
λ
(
1
+
λ
)
−
2
k
=
0
⇒
λ
2
+
λ
−
2
k
=
0
∵ Every square matrix satisfied its own characteristic equation.
∴
A
2
+
A
−
2
KI
=
0
⇒
A
2
=
2
KI
−
A
⇒
A
4
=
4
K
2
I
+
A
2
−
2
(
2
KI
)
(
A
)
⇒
A
4
=
4
K
2
I
+
2
KI
−
A
−
4
KA
⇒
A
4
=
(
4
K
2
+
2
K
)
I
−
(
1
+
4
K
)
A
...(i)
Now,
A
(
A
3
+
3
I
)
=
2
I
⇒
A
4
=
2
I
−
3
A
...(ii)
Comparing Eqs. (i) and (ii), we get
1
+
4
K
=
3
⇒
K
=
1
2
© examsnet.com
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