How to reduce non-digits or unknowns in a matrix?

Question

I have reduced this matrix to this point, but I am struggling on how to reduced it even more. If possible.

   1  0  -2    3
   0  1  1/a  -2/a
   0  0  a-1  a+2

Answer 1

These are the next steps:$$ \left[\begin{array}{ccc} 1 & 0 & -2 & 3 \\ 0 & 1 & \frac1a & -\frac2a \\ 0 & 0 & a-1 & a+2 \\ \end{array}\right]\to \left[\begin{array}{ccc} 1 & 0 & -2 & 3 \\ 0 & 1 & \frac1a & -\frac2a \\ 0 & 0 & \frac1a & {a+2 \over a(a-1)} \\ \end{array}\right] \to \left[\begin{array}{ccc} 1 & 0 & -2 & 3 \\ 0 & 1 & 0 & -{a+2 \over a(a-1)}-\frac2a \\ 0 & 0 & \frac1a & {a+2 \over a(a-1)} \\ \end{array}\right]\to \left[\begin{array}{ccc} 1 & 0 & 0 & 3+2{a+2 \over a-1} \\ 0 & 1 & 0 & -{a+2 \over a(a-1)}-\frac2a \\ 0 & 0 & 1 & {a+2 \over a-1} \\ \end{array}\right]= \left[\begin{array}{ccc} 1 & 0 & 0 & {5a+1 \over a-1} \\ 0 & 1 & 0 & -{3 \over a-1} \\ 0 & 0 & 1 & {a+2 \over a-1} \\ \end{array}\right] $$

If $a=0$, the matrix is undefined from the very begginning.

If $a=1$, the matrix is:

$$ \left[\begin{array}{ccc} 1 & 0 & -2 & 3 \\ 0 & 1 & 1 & -2 \\ 0 & 0 & 0 & 3 \\ \end{array}\right]\to \left[\begin{array}{ccc} 1 & 0 & -2 & 0 \\ 0 & 1 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ \end{array}\right] $$

Answer 2

If $a\ne 1$: \begin{align} \begin{bmatrix} 1&0&-2&3\\0&1&\frac1a&-\frac2a\\0&0&a-1&a+2 \end{bmatrix}&\rightsquigarrow \begin{bmatrix} 1&0&-2&3\\0&1&\frac1a&-\frac2a\\0&0&1&\frac{a+2}{a-1} \end{bmatrix}\rightsquigarrow \begin{bmatrix} 1&0&0&\frac{5a+1}{a-1}\\0&1&0&-\frac3{a-1}\\0&0&1&\frac{a+2}{a-1} \end{bmatrix}& \end{align}

If $a=1$, it is already reduced: $$ \begin{bmatrix} 1&0&-2&3\\0&1&1&-2\\0&0&0&3 \end{bmatrix}$$