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The system of equations in matrix form is as follows:

$\begin{bmatrix} 2 & 3 & -1 & 1 & 2 \\ 1 & 1 & -1 & 0 & 0 \\ 1 & 0 & 0 & 2 & -3 \\ 0 & -1 & 2 & -1 & 1 \\ 1 & -1 & 0 & 1 & -2 \end{bmatrix}\begin{bmatrix} a \\ b \\ c \\ d \\ e \end{bmatrix}=\begin{bmatrix} 9 \\ 4 \\ 4 \\ -6 \\ 0 \end{bmatrix}$

We can do the elimination (not Euler's here) to find $a, b, c, d$ and $e$ .

$\begin{matrix} R_1: \\ R_2: \\ R_3: \\ R_4: \\ R_5: \end{matrix} \begin{bmatrix} 2 & 3 & -1 & 1 & 2 & | & 9 \\ 1 & 1 & -1 & 0 & 0 & | & 4 \\ 1 & 0 & 0 & 2 & -3 & | & 4 \\ 0 & -1 & 2 & -1 & 1 & | & -6 \\ 1 & -1 & 0 & 1 & -2 & | & 0 \end{bmatrix}$

$\begin{matrix} R_1+R_5 &\rightarrow R_1: \\ R_3 &\rightarrow R_2: \\ R_3+3R_4 &\rightarrow R_3: \\ R_1-2R_4 &\rightarrow R_4: \\ R_5+2R_4 &\rightarrow R_5: \end{matrix} \begin{bmatrix} 3 & 2 & -1 & 2 & 0 & | & 9 \\ 1 & 1 & -1 & 0 & 0 & | & 4 \\ 1 & -3 & 6 & -1 & 0 & | & -14 \\ 2 & 5 & -5 & 3 & 0 & | & 21 \\ 1 & -3 & 4 & -1 & 0 & | & -12 \end{bmatrix}$

$\begin{matrix} R_1-3R_2 &\rightarrow R_1: \\ R_2-R_5 &\rightarrow R_2: \\ R_3-R_2 &\rightarrow R_3: \\ R_4-2R_2 & \rightarrow R_4: \end{matrix} \begin{bmatrix} 0 & -1 & 2 & 2 & 0 & | & -3 \\ 0 & 4 & -5 & 1 & 0 & | & 16 \\ 0 & -4 & 7 & -1 & 0 & | & -18 \\ 0 & 3 & -3 & 3 & 0 & | & 13 \end{bmatrix}$

$\begin{matrix} R_3+R_2 &\rightarrow R_1: \\ R_4+3R_1 &\rightarrow R_2: \\ R_1 &\rightarrow R_3: \\ \end{matrix} \begin{bmatrix} 0 & 0 & 2 & 0 & 0 & | & -2 \\ 0 & 0 & 3 & 9 & 0 & | & 4 \\ 0 & -1 & 2 & 2 & 0 & | & -3 \end{bmatrix}$

From $R_1: \quad 2 c = -2 \quad \Rightarrow c = -1$

Substituting $c = -1$ in $R_2: \quad 3(-1)+9d =4\quad \Rightarrow d = \frac {7}{9}$

Substituting $c = -1$ and $d = \frac {7}{9}$ in $R_2:$

$\quad \quad -b+2(-1)+2(\frac {7}{9}) = -3 \quad \Rightarrow b = \frac {23}{9}$

Substituting $b = \frac {23}{9}$ and $c = -1$ in $a+b-c=4 \quad \Rightarrow a = \frac {4}{9}$

Substituting the values of $b$ , $c$ and $d$ in $-b+2c-d+e=-6 \quad \Rightarrow e = -\frac {2}{3}$

Therefore, $\lceil 1000abcde \rceil = \lceil 1000 (\frac {4}{9} )( \frac {23}{9} )(-1) ( \frac {7}{9} )(-\frac {2}{3}) \rceil = \lceil 588.9346136 \rceil = \boxed {589}$

On solving, we will get $a=\frac{4}{9}, b=\frac{23}{9}, c=-1, d=\frac{7}{9}, e=\frac{-2}{3}$

abcde comes out to be 0.58893.... $\lceil{abcde}\rceil=1$ $1 \times 1000=\boxed{1000}$

What sharky did is first multiplied by 1000 and then used ceil function. So he is getting answer as 589.