  In this paper, we discussed the unique solvability of the two absolute value
matrix equations. The unique solvability condition $\rho (\vert A^{-1} B
\vert)<1$ is provided for the generalized absolute value matrix equation
(GAVME) $AX + B \vert X \vert = F$. This condition is superior to that of Kumar
et al. [J. Numer. Anal. Approx. Theory, 51(1) (2022) 83-87]. We also discussed
different conditions for the unique solvability of the new generalized absolute
value matrix equation (NGAVME) $AX+B\vert CX \vert=F$ with $A, B, C, F, X \in
\mathcal{R}^{n \times n}$. We also provided the corrected version of Corollary
2.1 from the published work by Wang et al. [Appl. Math. Lett., 116 (2021)
106966].
