Explain why the number of solutions $\mathbf{x} \in \mathbb{R}^{3}$ of the matrix equation $A \mathbf{x}=\mathbf{c}$ is 0,1 or infinity, where $A$ is a real $3 \times 3$ matrix and $\mathbf{c} \in \mathbb{R}^{3}$. State conditions on $A$ and $\mathbf{c}$ that distinguish between these possibilities, and state the relationship that holds between any two solutions when there are infinitely many.

Consider the case

$$A=\left(\begin{array}{lll} a & a & b \\ b & a & a \\ a & b & a \end{array}\right) \quad \text { and } \mathbf{c}=\left(\begin{array}{l} 1 \\ c \\ 1 \end{array}\right)$$

Use row and column operations to find and factorize the determinant of $A$.

Find the kernel and image of the linear map represented by $A$ for all values of $a$ and $b$. Find the general solution to $A \mathbf{x}=\mathbf{c}$ for all values of $a, b$ and $c$ for which a solution exists.