2
$\begingroup$

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 $\cos1,\cos2,\ldots,\cos n^2$. (For example,

$$d_3= \begin{vmatrix} \cos1&\cos2&\cos3\\ \cos4&\cos5&\cos6\\ \cos7&\cos8&\cos9\\ \end{vmatrix}$$

The argument of $\cos$ is always in radians, not degrees.) Evaluate $\lim_{n\to\infty}d_n$.

Interesting question, I don't know where to start.

$\endgroup$
1
$\begingroup$

$$\begin{array}{l} \left|\begin{array}{ccc}\cos 1 & \cos 2 & \cos 3 \\ \cos 4 & \cos 5 & \cos 6 \\ \cos 7 & \cos 8 & \cos 9 \end{array}\right| \\ =\left|\begin{array}{ccc}\cos 1 & \cos 2 & \cos 3 \\ \cos 3\cos 1 - \sin 3\sin 1 & \cos 3\cos 2 - \sin 3\sin 2 & \cos 3\cos 3 - \sin 3\sin 3 \\ \cos 7 & \cos 8 & \cos 9 \end{array}\right| \\= \left|\begin{array}{ccc}\cos 1 & \cos 2 & \cos 3 \\ - \sin 3\sin 1 & - \sin 3\sin 2 & - \sin 3\sin 3 \\ \cos 7 & \cos 8 & \cos 9 \end{array}\right|\\= -\sin 3 \left|\begin{array}{ccc}\cos 1 & \cos 2 & \cos 3 \\ \sin 1 & \sin 2 & \sin 3 \\ \cos 7 & \cos 8 & \cos 9 \end{array}\right| \\=-\sin 3 \left|\begin{array}{ccc}\cos 1 & \cos 2 & \cos 3 \\ \sin 1 & \sin 2 & \sin 3 \\ \cos 6\cos 1-\sin 6\sin 1 & \cos 6\cos 2-\sin 6\sin 2 & \cos 6\cos 3-\sin 6\sin 3 \end{array}\right| \\=-\sin 3 \left|\begin{array}{ccc}\cos 1 & \cos 2 & \cos 3 \\ \sin 1 & \sin 2 & \sin 3 \\ -\sin 6\sin 1 & -\sin 6\sin 2 & -\sin 6\sin 3 \end{array}\right| \\=\sin 3 \sin 6 \left|\begin{array}{ccc}\cos 1 & \cos 2 & \cos 3 \\ \sin 1 & \sin 2 & \sin 3 \\ \sin 1 & \sin 2 & \sin 3 \end{array}\right| =0. \end{array}$$

Can you get the value of $d_n$ following the same idea?

$\endgroup$
1
$\begingroup$

Hint: find a relation between $\cos(k)$, $\cos(k+1)$ and $\cos(k+2)$.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.