MathDB
Problems
Contests
Undergraduate contests
Putnam
1996 Putnam
4
Putnam 1996 B4
Putnam 1996 B4
Source:
June 6, 2014
Putnam
linear algebra
matrix
trigonometry
college contests
Problem Statement
For any square matrix
A
\mathcal{A}
A
we define
sin
A
\sin {\mathcal{A}}
sin
A
by the usual power series.
sin
A
=
∑
n
=
0
∞
(
−
1
)
n
(
2
n
+
1
)
!
A
2
n
+
1
\sin {\mathcal{A}}=\sum_{n=0}^{\infty}\frac{(-1)^n}{(2n+1)!}\mathcal{A}^{2n+1}
sin
A
=
n
=
0
∑
∞
(
2
n
+
1
)!
(
−
1
)
n
A
2
n
+
1
Prove or disprove :
∃
2
×
2
\exists 2\times 2
∃2
×
2
matrix
A
∈
M
2
(
R
)
A\in \mathcal{M}_2(\mathbb{R})
A
∈
M
2
(
R
)
such that
sin
A
=
(
1
1996
0
1
)
\sin{A}=\left(\begin{array}{cc}1 & 1996 \\0 & 1 \end{array}\right)
sin
A
=
(
1
0
1996
1
)
Back to Problems
View on AoPS