Answer
See below
Work Step by Step
Given $A=\begin{pmatrix}
1 & 2 & 3 & 4 & a\\
2 & 1 & 2 & 3 & 4\\
3 & 2 & 1 & 2 & 3 \\
4 & 3 & 2 & 1 & 2\\
a & 4 & 3 & 2 & 1
\end{pmatrix}$
By using software Mathemica, we get $\det(A)=-192+88a-8a^2=8(8-a)(a-3)\\
\rightarrow a=8, a=3$
Hence, $A$ is invertible if and only if $a\ne8, a\ne 3$
