The "matrice" in your example is ill - conditioned; thus, the computation of its inverse matrix is unstable (e.g., a small change in a matrix causes huge changes in the inverse matrix). I believe you already know the followings but just a reminder. If a matrix, X, is ill - conditioned, the inverse of the matrix is a "meaningless bad approximation". The condition number of the "matrice" in your example, (i.e., ratio of max singular value divided by min singular value), is gigantic (i.e., 2.6*10^10). Orthogonal matrix should have singular value of one.
Ill-conditioning is also known as multicollinearity in regression, confounding (or alias structure) in design of experiments, rank-deficient in matrix algebra. Note that the inverse of an orthogonal matrix is the same as its transpose (i.e., the U and V are orthogonal matrices, U'U=I, V'V=I.
X=U*S*V'.
I always compute Singular Value Decomposition (SVD) if a matrix computation is involved because SVD is the most reliable way to compute an inverse matrix. I checked Mathematica commands such as PseudoInverse, Inverse, LinearSolve, LeastSquares for ill-conditioning in the past and these commands do not provide a warning sometimes. Matlab has also the same issue.
{u, s, v} = SingularValueDecomposition[matrice]
ListPlot[Diagonal[s], PlotRange -> {All, {0, 60}}]
condNumber=Max[Diagonal[s]]/Min[Diagonal[s]] (* 2.6*10^10 *)
