Tesler matrices are certain integral matrices counted by the Kostant partition function and have appeared recently in Haglund's study of diagonal harmonics. In 2014, Drew Armstrong defined a poset on such matrices and conjectured that the characteristic polynomial of this poset is a power of $(q-1)$. We will use a method of Bruce Sagan and Joshua Hallam to prove Armstrong's conjecture and explore how this result can improve the bounds on the number of Tesler matrices.