I have to compute the following differentiation :

\(\frac{\partial}{\partial\sigma^2}\det[\mathbf X_{p\times n}'(\sigma^2 \mathbf I_{n}+\mathbf Z_{n\times q}\mathbf G_{q\times q}\mathbf Z_{q\times n}')^{-1}\mathbf X_{n\times p}],\)

where \(\sigma^2\) is a scalar, \(\det\) denotes determinant, \(\mathbf I_{n}\) is a $n\times n$ identity matrix. Note that, \(\mathbf X\), \(\mathbf Z\), and \(\mathbf G\) do NOT involve \(\sigma^2\).

How can I do that?

\(\frac{\partial}{\partial\sigma^2}\det[\mathbf X_{p\times n}'(\sigma^2 \mathbf I_{n}+\mathbf Z_{n\times q}\mathbf G_{q\times q}\mathbf Z_{q\times n}')^{-1}\mathbf X_{n\times p}],\)

where \(\sigma^2\) is a scalar, \(\det\) denotes determinant, \(\mathbf I_{n}\) is a $n\times n$ identity matrix. Note that, \(\mathbf X\), \(\mathbf Z\), and \(\mathbf G\) do NOT involve \(\sigma^2\).

How can I do that?

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