Putnam

Algebra Difficulty 7.8 National Olympiad, round 2 Find the answer

Let $d_n$ be the determinant of the $n \times n$ matrix whose entries, from
left to right and then from top to bottom, are $\cos 1, \cos 2, \dots, \cos n^2$ . Evaluate
$\lim_{n\to\infty} d_n$ .

Your answer

A number or a short expression. Fractions can be typed as 3/2, and spacing doesn't matter.

Solution

The limit is $0$ ; we will show this by checking that $d_n = 0$ for all $n \geq 3$ .
Starting from the given matrix, add the third column to the first column; this does not change the
determinant. However, thanks to the identity
$\cos x + \cos y = 2 \cos \frac{x+y}{2} \cos \frac{x-y}{2}$ ,
the resulting matrix has the form
$\begin{pmatrix} 2 \cos 2 \cos 1 & \cos 2 & \cdots \\ 2 \cos (n+2) \cos 1 & \cos (n+2) & \cdots \\ 2 \cos (2n+2) \cos 1 & 2 \cos (2n+2) & \cdots \\ \vdots & \vdots & \ddots \end{pmatrix}$
with the first column being a multiple of the second. Hence $d_n = 0$ .

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