I want to reformulate the following determinant: $$ \left| \left(A^TB^{-1}A+ \frac{1}{c}C^{-1} \right)^{-1} \right|^{1/2} $$
where $c $ is a real scalar, $A$ is an ($n\times k$) matrix, $B$ is an ($n\times n$) matrix and C is a ($k\times k$) matrix. The dimension of matrix inside the above determinant is $k\times k$, so it is a square matrix.
By using the property of $\det(A^{-1})=\det(A)^{-1}$, the expression is written as
$$ \left|A^TB^{-1}A+ \frac{1}{c}C^{-1} \right|^{-1/2} $$
How to reformulate this expression further?
I have read somewhere that it is equivalent to $$ \left| \frac{1}{c}C^{-1} \right|^{-1/2} \left| B \right|^{1/2} \left| B+cACA^T \right|^{-1/2}. $$
Is this correct?