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Joint Entrance Examination
Mathematics
Matrices and Determinants
Question 271
Question 271 of 472
Medium
MCQ
If
P
=
(
1
0
1
/
2
1
)
P=\begin{pmatrix}1&0\\1/2&1\end{pmatrix}
P
=
(
1
1/2
0
1
)
, then
P
50
P^{50}
P
50
is
A
(
1
&
0
25
&
1
)
\begin{pmatrix}1\&0\\25\&1\end{pmatrix}
(
1&0
25&1
)
B
(
1
&
50
0
&
1
)
\begin{pmatrix}1\&50\\0\&1\end{pmatrix}
(
1&50
0&1
)
C
(
1
&
25
0
&
1
)
\begin{pmatrix}1\&25\\0\&1\end{pmatrix}
(
1&25
0&1
)
D
(
1
&
0
50
&
1
)
\begin{pmatrix}1\&0\\50\&1\end{pmatrix}
(
1&0
50&1
)
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Solutions · 1
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Official solution
By induction,
P
n
=
(
1
0
n
/
2
1
)
P^n=\begin{pmatrix}1&0\\n/2&1\end{pmatrix}
P
n
=
(
1
n
/2
0
1
)
(a lower-triangular unipotent pattern).
P
50
=
(
1
0
25
1
)
P^{50}=\begin{pmatrix}1&0\\25&1\end{pmatrix}
P
50
=
(
1
25
0
1
)
.
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