$$
\newcommand{\uex}{{u_{\small\mbox{e}}}}
\newcommand{\Aex}{{A_{\small\mbox{e}}}}
\newcommand{\half}{\frac{1}{2}}
\newcommand{\tp}{\thinspace .}
\newcommand{\Oof}[1]{\mathcal{O}(#1)}
\newcommand{\x}{\boldsymbol{x}}
\newcommand{\X}{\boldsymbol{X}}
\renewcommand{\u}{\boldsymbol{u}}
\renewcommand{\v}{\boldsymbol{v}}
\newcommand{\e}{\boldsymbol{e}}
\newcommand{\f}{\boldsymbol{f}}
\newcommand{\dfc}{\alpha} % diffusion coefficient
\newcommand{\Ix}{\mathcal{I}_x}
\newcommand{\Iy}{\mathcal{I}_y}
\newcommand{\Iz}{\mathcal{I}_z}
\newcommand{\If}{\mathcal{I}_s} % for FEM
\newcommand{\Ifd}{{I_d}} % for FEM
\newcommand{\Ifb}{{I_b}} % for FEM
\newcommand{\sequencei}[1]{\left\{ {#1}_i \right\}_{i\in\If}}
\newcommand{\basphi}{\varphi}
\newcommand{\baspsi}{\psi}
\newcommand{\refphi}{\tilde\basphi}
\newcommand{\psib}{\boldsymbol{\psi}}
\newcommand{\sinL}[1]{\sin\left((#1+1)\pi\frac{x}{L}\right)}
\newcommand{\xno}[1]{x_{#1}}
\newcommand{\Xno}[1]{X_{(#1)}}
\newcommand{\xdno}[1]{\boldsymbol{x}_{#1}}
\newcommand{\dX}{\, \mathrm{d}X}
\newcommand{\dx}{\, \mathrm{d}x}
\newcommand{\ds}{\, \mathrm{d}s}
$$
Calculating a general row in the matrix; details
$$
\begin{align*}
A_{i,i-1} &= \int_\Omega \basphi_i\basphi_{i-1}\dx\\
&=
\underbrace{\int_{\xno{i-2}}^{\xno{i-1}} \basphi_i\basphi_{i-1}\dx}_{\basphi_i=0} +
\int_{\xno{i-1}}^{\xno{i}} \basphi_i\basphi_{i-1}\dx +
\underbrace{\int_{\xno{i}}^{\xno{i+1}} \basphi_i\basphi_{i-1}\dx}_{\basphi_{i-1}=0}\\
&= \int_{\xno{i-1}}^{\xno{i}}
\underbrace{\left(\frac{x - x_{i}}{h}\right)}_{\basphi_i(x)}
\underbrace{\left(1 - \frac{x - \xno{i-1}}{h}\right)}_{\basphi_{i-1}(x)} \dx =
\frac{h}{6}
\end{align*}
$$
- \( A_{i,i+1}=A_{i,i-1} \) due to symmetry
- \( A_{i,i}=2h/3 \) (same calculation as for \( A_{2,2} \))
- \( A_{0,0}=A_{N,N}=h/3 \) (only one element)
