$$(\boldsymbol{A}, \boldsymbol{b})=\left(\begin{array}{ccccc} 1 & 1 & 1 & 1 & 1 \\ 3 & 2 & 1 & 1 & 0 \\ 0 & 1 & 2 & 2 & 3 \\ 5 & 4 & 3 & 3 & a \end{array}\right)$$
$\sim$
$$\left(\begin{array}{rrrrr} 1 & 1 & 1 & 1 & 1 \\ 0 & -1 & -2 & -2 & -3 \\ 0 & 1 & 2 & 2 & 3 \\ 0 & -1 & -2 & -2 & a-5 \end{array}\right)$$
$\sim$
$$\left(\begin{array}{rrrrr} 1 & 1 & 1 & 1 & 1 \\ 0 & 1 & 2 & 2 & 3 \\ 0 & 1 & 2 & 2 & 3 \\ 0 & -1 & -2 & -2 & a-5 \end{array}\right)$$

$\sim$

$$\left(\begin{array}{rrrrr} 1 & 0 & -1 & -1 & -2 \\ 0 & 1 & 2 & 2 & 3 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & a-2 \end{array}\right)$$
$\sim$
$$\left(\begin{array}{rrrrr} x_1 & x_2 & x_3 & x_4 & b \\ 1 & 0 & -1 & -1 & -2 \\ 0 & 1 & 2 & 2 & 3 \\ 0 & 0 & 0 & 0 & a-2 \\ 0 & 0 & 0 & 0 & 0 \end{array}\right)$$

(1)当 $a-2 \neq 0$, 即 $a \neq 2$ 时, $\mathrm{r}(\boldsymbol{A})=2, \mathrm{r}(\boldsymbol{A}, \boldsymbol{b})=3, \mathrm{r}(\boldsymbol{A}) \neq \mathrm{r}(\boldsymbol{A}, \boldsymbol{b})$, 线性方程组无解;

(2) 当 $a-2=0$, 即 $a=2$ 时, $\mathrm{r}(\boldsymbol{A})=2, \mathrm{r}(\boldsymbol{A}, \boldsymbol{b})=2, \mathrm{r}(\boldsymbol{A})=\mathrm{r}(\boldsymbol{A}, \boldsymbol{b}) < n=4$, 线性方程组有无穷多解;

$$\boldsymbol{\xi}_1=\left(\begin{array}{r} 1 \\ -2 \\ 1 \\ 0 \end{array}\right), \boldsymbol{\xi}_2=\left(\begin{array}{r} 1 \\ -2 \\ 0 \\ 1 \end{array}\right),$$

$$\boldsymbol{x}=\boldsymbol{\eta}^*+c_1 \boldsymbol{\xi}_1+c_2 \boldsymbol{\xi}_2=\left(\begin{array}{r} -2 \\ 3 \\ 0 \\ 0 \end{array}\right)+c_1\left(\begin{array}{r} 1 \\ -2 \\ 1 \\ 0 \end{array}\right)+c_2\left(\begin{array}{r} 1 \\ -2 \\ 0 \\ 1 \end{array}\right),\left(c_1, c_2 \in \mathbb{R}\right)$$