A determinant (2)
Posted: Tue Jan 19, 2016 3:13 pm
Let $[a, b]$ denote the least common multiple of $a, b$. Prove that:
$$\begin{vmatrix}
\left [ 1,1 \right ] &\left [ 1,2 \right ] &\cdots &\left [1,n \right ] \\
\left [ 2,1 \right ]&\left [ 2,2 \right ] & \cdots &\left [ 2,n \right ] \\
\vdots & \vdots & \ddots & \vdots \\
\left [ n,1 \right ]& \left [ n, 2 \right ] &\cdots & \left [ n,n \right ]
\end{vmatrix} = \prod_{k=1}^{n}\phi(k) \prod_{p \mid k} (-p)$$
where $\phi$ denotes Euler's $\phi$ function.
$$\begin{vmatrix}
\left [ 1,1 \right ] &\left [ 1,2 \right ] &\cdots &\left [1,n \right ] \\
\left [ 2,1 \right ]&\left [ 2,2 \right ] & \cdots &\left [ 2,n \right ] \\
\vdots & \vdots & \ddots & \vdots \\
\left [ n,1 \right ]& \left [ n, 2 \right ] &\cdots & \left [ n,n \right ]
\end{vmatrix} = \prod_{k=1}^{n}\phi(k) \prod_{p \mid k} (-p)$$
where $\phi$ denotes Euler's $\phi$ function.