A determinant
Posted: Thu Mar 24, 2016 8:12 am
This post rang me also bells about a famous determinant.
Let $\gcd(i,j)$ denote the greatest common divisor of $i,j$ and let $\varphi$ denote Euler's totient function. Prove that:
$$\begin{vmatrix}
\gcd(1,1) &\gcd(1, 2) &\cdots & \gcd(1,n)\\
\gcd(2,1)&\gcd(2,2) &\cdots & \gcd(2,n)\\
\vdots& \vdots & \ddots &\vdots \\
\gcd(n,1)&\gcd(n,2) &\cdots &\gcd(n,n)
\end{vmatrix}= \prod_{j=1}^{n}\varphi(j)$$
Edit: Topic moved from "Linear Algebra" to "General Mathematics".
Let $\gcd(i,j)$ denote the greatest common divisor of $i,j$ and let $\varphi$ denote Euler's totient function. Prove that:
$$\begin{vmatrix}
\gcd(1,1) &\gcd(1, 2) &\cdots & \gcd(1,n)\\
\gcd(2,1)&\gcd(2,2) &\cdots & \gcd(2,n)\\
\vdots& \vdots & \ddots &\vdots \\
\gcd(n,1)&\gcd(n,2) &\cdots &\gcd(n,n)
\end{vmatrix}= \prod_{j=1}^{n}\varphi(j)$$
Edit: Topic moved from "Linear Algebra" to "General Mathematics".