Explain why the number of solutions $\mathbf{x}$ of the simultaneous linear equations $A \mathbf{x}=\mathbf{b}$ is 0,1 or infinity, where $A$ is a real $3 \times 3$ matrix and $\mathbf{x}$ and $\mathbf{b}$ are vectors in $\mathbb{R}^{3}$. State necessary and sufficient conditions on $A$ and $\mathrm{b}$ for each of these possibilities to hold.

Let $A$ and $B$ be real $3 \times 3$ matrices. Give necessary and sufficient conditions on $A$ for there to exist a unique real $3 \times 3$ matrix $X$ satisfying $A X=B$.

Find $X$ when

$$A=\left(\begin{array}{ccc} 1 & 1 & 2 \\ 1 & 0 & 1 \\ 1 & 2 & 0 \end{array}\right) \quad \text { and } \quad B=\left(\begin{array}{ccc} 4 & 0 & 1 \\ 2 & 1 & 0 \\ 3 & -1 & -1 \end{array}\right)$$