Rank of a real matrix

Question

Let $T$ be a noninvertible matrix in $M_n(\mathbb R)$ and $L$ be a one-rank matrix on $T\mathcal R^n$. Now, consider $TRT=LT$ on $\mathcal R^n$ where $R$ is a matrix in $M_n(\mathbb R)$.

Is the rank of $R$ one? Can we get $R=T_{1}^{-1}LT$ where $T_{1}^{-1}$ is inverse of the restriction of $T$ on $T\mathcal R^n$?

Answer 1

The answer is no. For example, take $$ T = \pmatrix{1\\ &1 \\&&0}, \quad L = \pmatrix{1\\&0\\&&0}, \quad R = \pmatrix{1\\ &0\\ &&1} $$ and note that $TR = L$. Notably, if $T$ has rank $1$, then $R$ might have any rank.