<s>[INST] # Joint likelihood of the non-centrality parameters of a multivariate normal distribution

I am trying to arrive at the joint likelihood of the non-centrality parameters of a multivariate normal distribution. I am using the following definition of the multivariate normal distribution:

$$
\text{Let $\mathbf{X}$ be a $k \times m$ random matrix with rows $\mathbf{X}_1^T, \mathbf{X}_2^T, ..., \mathbf{X}_m^T$. Suppose that each row $\mathbf{X}_i^T$ has a multivariate normal distribution with mean vector $\boldsymbol{\mu}_i$ and covariance matrix $\Sigma$. Suppose also that the rows are independent. The joint density function of the rows is then given by
$$
(2\pi)^{-\frac{mk}{2}} |\Sigma|^{-\frac{m}{2}}  \text{etr}\left(-\frac{1}{2}(\mathbf{X}-\mathbf{M})^T \Sigma^{-1}(\mathbf{X}-\mathbf{M})\right)
$$
}
$$
where $\mathbf{M}$ is the $k \times m$ matrix with rows $\boldsymbol{\mu}_1^T, \boldsymbol{\mu}_2^T, ..., \boldsymbol{\mu}_m^T$, and the function $\text{etr}$ is the exponential trace function given by
$\text{etr}(\mathbf{A}) = \exp(\text{tr} \mathbf{A}).$

After applying the trace properties that (\text{tr}(\mathbf{A}+\mathbf{B}) = \text{tr}(\mathbf{A}) + \text{tr}(\mathbf{B})$ and $\text{tr}(\mathbf{AB})=\text{tr}(\mathbf{BA})$, I arrived at the expression:
$$
L(\mathbf{M}) = (2\pi)^{-\frac{mk}{2}} |\Sigma|^{-\frac{m}{2}}  \exp \left(-\frac{1}{2}\left(-2\text{tr}(\Sigma^{-1}\mathbf{M}^T \mathbf{X}) + \text{tr}(\Sigma^{-1}\mathbf{M}^T \mathbf{M}) + \text{tr}(\Sigma^{-1}\mathbf{X}^T \mathbf{X})\right)\right)
$$

However, it is given in a paper that the joint likelihood of non-centrality parameters ($\mathbf{M}$)  are
$$
L({\mathbf{M}}) = (2\pi)^{\frac{-mk}{2}} |\Sigma|^{-\frac{m}{2}} \exp \left( -\frac{1}{2} \text{tr} (\mathbf{X}\mathbf{X}^T - 2\mathbf{X}\mathbf{M}^T + \mathbf{M}\mathbf{M}^T)\Sigma^{-1} \right).
$$

The source of the discrepancy seems to be that I assumed the wrong properties of the trace function. What properties should I apply?[/INST]