Let $d' < d$ be natural numbers, and let $L$ be a real matrix of dimension $d'\times d$, with maximum rank (that is, with rank $d'$). Let $A$ be a real square matrix of dimension $d$, and let's suppose that $A$ is regular. We consider the product $LAL^T$. This product is a square matrix of dimension $d'$ and I want to see that it is also regular, or equivalently, that it has rank $d'$.
EDIT. Thanks to the answers now I see that the regular condition for $A$ is not enough, but what about if $A$ is positive definite?
I have tried to see the matrices as linear maps and use the rank-nullity theorem to obtain information, but I'm only able to find that the rank of $AL^T$ is $d'$. Any ideas?
Also I'm interested in knowing if the same follows if we replace $L^T$ by any other matrix of maximum rank with the same dimensions. That is, the product $LAM$ will be regular for matrices $L$ and $M$ with dimensions $d'\times d$ and $d\times d'$, respectively, and both ranks $d'$?