Let's write out the equations for $N_x=3$

Equation (10) written for $i=1,\ldots,Nx-1= 1,2$ becomes $$\begin{align} \frac{u^{n}_1-u^{n-1}_1}{\Delta t} &= \dfc\frac{u^{n}_{2} - 2u^n_1 + u^n_{0}}{\Delta x^2}\\ \frac{u^{n}_2-u^{n-1}_2}{\Delta t} &= \dfc\frac{u^{n}_{3} - 2u^n_2 + u^n_{1}}{\Delta x^2} \end{align}$$

(The boundary values $u^n_0$ and $u^n_3$ are known as zero.)

Collecting the unknown new values on the left-hand side and writing as $2\times 2$ matrix system: $$\left(\begin{array}{cc} 1+ 2F & - F\\ - F & 1+ 2F \end{array}\right) \left(\begin{array}{c} u^{n}_1\\ u^{n}_{2}\\ \end{array}\right) = \left(\begin{array}{c} u^{n-1}_1\\ u^{n-1}_2 \end{array}\right)$$