The vector $\mathbf{x}=\left(\begin{array}{l}x \\ y \\ z\end{array}\right)$ satisfies the equation

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

where $\mathbf{A}$ is a $(3 \times 3)$ matrix and $\mathbf{b}$ is a $(3 \times 1)$ column vector. State the conditions under which this equation has (a) a unique solution, (b) an infinity of solutions, (c) no solution for $\mathbf{x}$.

Find all possible solutions for the unknowns $x, y$ and $z$ which satisfy the following equations:

$$\begin{array}{r} x+y+z=1 \\ x+y+\lambda z=2 \\ x+2 y+\lambda z=4 \end{array}$$

in the cases (a) $\lambda=0$, and (b) $\lambda=1$.