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Joint Entrance Examination
Mathematics
Matrices and Determinants
Question 304
Question 304 of 472
Medium
MCQ
If
ω
≠
1
\omega\ne1
ω
=
1
is the complex cube root of unity and matrix
H
=
(
ω
0
0
ω
)
H=\begin{pmatrix}\omega&0\\0&\omega\end{pmatrix}
H
=
(
ω
0
0
ω
)
, then
H
70
H^{70}
H
70
is equal to
A
0
B
−
H
-H
−
H
C
H
2
H^2
H
2
D
H
H
H
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Official solution
H
k
=
(
ω
k
0
0
ω
k
)
H^k=\begin{pmatrix}\omega^k&0\\0&\omega^k\end{pmatrix}
H
k
=
(
ω
k
0
0
ω
k
)
. Since
ω
3
=
1
\omega^3=1
ω
3
=
1
:
H
70
=
H
69
⋅
H
=
(
ω
69
)
H
=
(
ω
3
)
23
H
=
H
H^{70}=H^{69}\cdot H=(\omega^{69})H=(\omega^3)^{23}H=H
H
70
=
H
69
⋅
H
=
(
ω
69
)
H
=
(
ω
3
)
23
H
=
H
.
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