What is this "1" in this stiffness matrix?
$$K_e= \frac{E^e A^e 1}{L^e} \begin{bmatrix}1&-1\\1&-1\end{bmatrix}$$
eq. 6, p. 2.
Typo or some kind of indicator perhaps? I don't see it significant, because even without it, it expresses the system of eqs 3 and 4.
I think so yes, the $E^eA^e1=E^eA^e$. But, I found another mistake. The matrix should be \begin{bmatrix} 1 &-1\\ -1 & 1 \end{bmatrix} Your $2^{\mathrm{nd}}$ equation has $-1$ with $u_1$.