Matrix inversion with variable in {-1,1}

Question

Could you please give me a hint for computing inversion of this matrix?

$$ \begin{pmatrix} 1 & f & g+h\sqrt(2) \\ 0 & i & j \\ 0 & 0 & 1 \\ \end{pmatrix} $$

where $f,j \in \mathbb Z;g,h\in \mathbb Q;i\in \{-1,1\}$

I can't use this formula:

$$ \displaystyle (A^{-1})_{ij}=(-1)^{i+j}\frac{\mathop{\rm det}\nolimits A_{j,i}}{\mathop{\rm det}\nolimits A}\, $$

I'm getting this matrix: $$ \begin{pmatrix} 1 & f & 0 \\ 0 & i & j \\ 0 & 0 & 1 \\ \end{pmatrix} $$

but don't know what to do next, as $i$ is $\{1,-1\}$

Answer 1

$$ \begin{pmatrix} 1 & f & g+h\sqrt(2) \\ 0 & i & j \\ 0 & 0 & 1 \\ \end{pmatrix}^{-1}= \frac1i \begin{pmatrix} i & -f & \;\;fj-ig-ih\sqrt(2) \\ 0 & \;\;1 & \!-j \\ 0 & \;\;0 & \;\;i \\ \end{pmatrix} $$

Answer 2

Done using augmented matrix: $\left(\begin{array}{ccc}1&-fi&-g-h\sqrt2+fij\\0&i&-ij\\0&0&1\end{array}\right)$