  Let $\mathbf{B}_n=\mathbf {S}_n(\mathbf {S}_n+\alpha_n\mathbf {T}_N)^{-1}$,
where $\mathbf {S}_n$ and $\mathbf {T}_N$ are two independent sample covariance
matrices with dimension $p$ and sample sizes $n$ and $N$, respectively. This is
the so-called Beta matrix. In this paper, we focus on the limiting spectral
distribution function and the central limit theorem of linear spectral
statistics of $\mathbf {B}_n$. Especially, we do not require $\mathbf {S}_n$ or
$\mathbf {T}_N$ to be invertible. Namely, we can deal with the case where
$p>\max\{n,N\}$ and $p<n+N$. Therefore, our results cover many important
applications which cannot be simply deduced from the corresponding results for
multivariate $F$ matrices.
