I'm trying to come up with a method to generate a random inverbile matrix. This is my guess.
Assume $A \in \mathbb{R}^{n\times n}$, such that the $LU$ decomposition can be applied. Therefore we'll have
$$ \begin{array}{l} A = LU \\ (L)_{ij} = \begin{cases} 0 & if \; i<j \\ 1 & if \; i = j \\ l_{ij} \in \mathbb{R} &if\;i>j \end{cases} \\ (U)_{ij} = \begin{cases} u_{ij} \in \mathbb{R} & if \; i < j \\ u_{ij} \in \mathbb{R}-\left\{ 0 \right\} & if \; i = j \\ 0 &if\;i>j \end{cases} \end{array}, $$ therefore I need to generate all the $l_{ij},u_{ij}$ according to the constraints above. If those constraints are met then
$$ det(A) = det(L)det(U) = det(U) = \prod_{i=1}^{n}u_{ii} $$
I guess this method might be valid, however I don't think that the distrubution of $A$ will be uniform across $R \subset \mathbb{R}^{n\times n}$. For example if $l_{11}$ and $u_{11}$ are uniformly distributed in some compact subset of $\mathbb{R}$ then the distribution of $a_{11}$ won't be uniform.
Is the method valid? I think this can be easily implemented on a computer. Are there methods well know in literature for this?
Here are some possibilities.
A Householder reflector $$ Q = I - vv^T, \quad v \in \mathbb{R}^n, \quad v^Tv = 2$$ is an orthogonal matrix. In particular, $Q$ is nonsingular with condition number $1$. This option gives you a cheap inverse.
If $V$ is orthogonal and $\Lambda$ is real and diagonal, then $$ A = V \Lambda V^T$$ is symmetric. If $\Lambda > 0$, then $A$ is symmetric positive definite and $A$ has an $LU$ decomposition. The inverse is simple and you have control over the condition number.
If the matrices $\{B_i\}_{i=1}^m$ are block diagonal, then their product $B = B_1B_2\dots B_m$ is banded. This is true in general, even when the block structures are not identical. If the matrices $B_i$ are nonsingular, then the matrix $B^{-1}$ is also banded. This gives you an cheap inverse.
If $Q$ is orthogonal and $S$ is quasi-upper-triangular, with either 1 by 1 or 2 by 2 blocks on the main diagonal, then $$ A = Q S Q^T $$ is a general dense matrix, where you have control over the (possibly) complex spectrum.