Change of variable (Jacobian) with matrix-variate distributions

Question

I'm working with matrix-variate distributions and MCMC sampling. What i'm doing is imposing a prior probability distribution over a matrix $\Lambda$, but since this is a covariance matrix (hence PSD) i'm working with it's Cholesky decomposition $\Lambda = LL^T$. Thus, i'm performing MCMC sampling of the matrix $L$ but imposing a probability distribution over $\Lambda$. This implies a change of variable so the adjustment of the Jacobian is needed: $$ p(L)dL = p(\Lambda)d\Lambda \rightarrow p(L) = p(\Lambda)\left|\frac{dvec(\Lambda)}{dvec(L)}\right| = p(\Lambda)|J(\Lambda,L)| $$ I found here that the Jacobian of the transformation $Y = TT'$ (with $T$ lower triangular) is $2^n\prod_i{t_{ii}^{n-i+1}}$ and since i work with log-probabilities what i get is: $$ \log{p(L)} = \log{p(\Lambda)} + \log{2^n\prod_i{t_{ii}^{n-i+1}}} $$ I tried an experiment with the Wishart distribution (sampling $L$ but imposing a Wishart on $\Lambda$) and seems working but i tried one other experiment with a matrix-variate normal distribution over $\Lambda$ (and sampling $L$) and it's not working well for the diagonal elements... Now the question: is this reasoning correct when working with matrix-variate distributions?