Daily math problems for May 14, 2026 — Matrices, Limits, Complex Numbers

$\det A=0$, so $A$ is singular (not invertible).

${\displaystyle \frac{\sin 3x}{\tan 5x}}={\displaystyle \frac{\sin 3x}{3x}}\cdot {\displaystyle \frac{5x}{\tan 5x}}\cdot {\displaystyle \frac{3}{5}}\to 1\cdot 1\cdot {\displaystyle \frac{3}{5}}={\displaystyle \frac{3}{5}}$.

The polynomial $z^n-1$ has zero coefficient for $z^{n-1}$, so by Viète's formula the sum of its roots is $0$. Equivalently, the geometric sum ${\sum }_{k=0}^{n-1}{e}^{2\pi ik\mathrm{/}n}={\displaystyle \frac{1-1}{1-e^{2\pi i\mathrm{/}n}}}=0$ (for $n\ge 2$).