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We only know that $B$ is a $n\times n$ singular matrix and $D$ is a $n\times n$ non-singular matrix. We have no idea singularity for $A$ and $C$.

If $AB=CD^{-1}$ holds, do we get any information about singularity for $A$ and/or $C$?

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    $\det(AB)=\det(A)\det(B)$ and $\det(CD^{-1})=\det(C)\det(D^{-1})$. Now, what does singularity have to do with determinant? What is $\det(B)$ equal to? what does that mean $\det(AB)$ is equal to? What does that mean $\det(CD^{-1})$ is equal to? Finally, what does this mean $\det(C)$ is equal to? – 2017-02-21
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    As for tag usage, please *read* what the tag is about before using it. Singularity theory has nothing to do with this question. – 2017-02-21

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I'm presuming $A$ and $C$ are also $n \times n$. Since $B$ is singular, $AB$ is also singular, so $C = ABD$ must be singular. But $A$ could be anything.

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    It is greatly appreciated. – 2017-02-22