Year: 2009
Topic: Linear Algebra Trigonometry

Problem:
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 $\cos 1, \cos 2, \dots, \cos n^2$. (For example, \[  d_3 = \left| \begin{matrix} \cos 1 & \cos 2 & \cos 3 \\                \cos 4 & \cos 5 & \cos 6 \\ \cos 7 & \cos 8 & \cos 9               \end{matrix} \right|. \] The argument of $\cos$ is always in radians, not degrees.) Evaluate $\lim_{n\to\infty} d_n$.

Solution:
We will start 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 $d_n = 0$. The limit is $\boxed{0}$.