Let \(\pi (A)\) , \(\xi (A)\) , and \(\nu (A)\) , respectively, denote the number of positive, zero, and negative eigenvalues of the matrix A. Then the triplet \((\pi (A), \xi (A), \nu (A))\) is called the inertia of A and is denoted by \(\textrm{Inertia}(A)\) . Let \(\beta \) be the beta function. The inertia of the matrix \([\beta (i,j )]\) is shown to be \(\left( \frac{n}{2},0,\frac{n}{2}\right) \) if n is even, and \(\left( \frac{n+1}{2},0,\frac{n-1}{2}\right) \) if n is odd. It is also shown that \([\beta (i,j)]\) is Birkhoff–James orthogonal to the \(n\times n\) identity matrix I in the trace norm if and only if n is even. For \(0<{\lambda }_1<\cdots<{\lambda }_n, 0<\mu _1<\cdots <\mu _n\) , it is shown that the matrix \([(\beta ({\lambda }_i,\mu _j))^m]\) is non singular if \(\mu _{i+1}-\mu _{i}\in {{\mathbb {N}}}\) for all \(1\le i \le n-1\) . It is also shown that if \(\mu _{i+1}-\mu _i \in {{\mathbb {N}}}\) for \(1\le i\le n-1\) , then for \(m\in {\mathbb {N}}\) , the matrix \(\left[ \frac{1}{\beta ({\lambda }_i,\mu _j)^m}\right] \) is totally positive.
