Let $\mathbf{A}$ be a real $3 \times 3$ matrix. Define the rank of $\mathbf{A}$. Describe the space of solutions of the equation

$$\mathbf{A x}=\mathbf{b},$$

organizing your discussion with reference to the rank of $\mathbf{A}$.

Write down the equation of the tangent plane at $(0,1,1)$ on the sphere $x_{1}^{2}+x_{2}^{2}+x_{3}^{2}=2$ and the equation of a general line in $\mathbb{R}^{3}$ passing through the origin $(0,0,0)$.

Express the problem of finding points on the intersection of the tangent plane and the line in the form $(\dagger)$. Find, and give geometrical interpretations of, the solutions.