  In this paper we give alternate proofs of some well-known matrix
inequalities. In particular, we show that under certain conditions the
inequality holds \begin{align}\sum \limits_{\lambda_i\in
\mathrm{Spec}(ab^{T})}\mathrm{min}\{\log |t-\lambda_i|\}_{[||a||,||b||]}&\leq
\# \mathrm{Spec}(ab^T)\log\bigg(\frac{||b||+||a||}{2}\bigg)\nonumber
\\&+\frac{1}{||b||-||a||}\sum \limits_{\lambda_i\in \mathrm{Spec}(ab^T)}\log
\bigg(1-\frac{2\lambda_i}{||b||+||a||}\bigg).\nonumber \end{align}Also under
the same condition, the inequality also holds\begin{align}\int
\limits_{||a||}^{||b||}\log|\mathrm{det}(ab^{T}-tI)|dt&\leq \#
\mathrm{Spec}(ab^T)(||b||-||a||)\log\bigg(\frac{||b||+||a||}{2}\bigg)\nonumber
\\&+\sum \limits_{\lambda_i\in \mathrm{Spec}(ab^T)}\log
\bigg(1-\frac{2\lambda_i}{||b||+||a||}\bigg).\nonumber \end{align}
