Q1: Let $A, T \in \mathbb{F}^{n \times n}$ and $T$ be invertible. What is always correct?

A1: $\mathrm{Ker}(T A) = \mathrm{Ker}(A)$

A2: $\mathrm{Ker}( A T) = \mathrm{Ker}(A)$

A3: $\mathrm{Ran}(T A) = \mathrm{Ran}(A)$

A4: $\mathrm{Ker}(T A) = \mathrm{Ran}(A T)$

Q2: Let $A, T, S \in \mathbb{F}^{n \times n}$ and $S, T$ be invertible. What is always correct?

A1: $\mathrm{rank}(SAT) = \mathrm{rank}(A)$

A2: $\mathrm{rank}(SAT) = n$

A3: $\mathrm{rank}(SAT) = \mathrm{rank}(T)$

A4: $\mathrm{rank}(A) + \mathrm{dim} \mathrm{Ker}(T) = n$

Q3: Are the matrices $A = \begin{pmatrix} 1 & 0 & 1 \ 0 & 0 & 1 \end{pmatrix}$ and $B = \begin{pmatrix} 1 & 1 & 0\ 1 & 1 & 0 \end{pmatrix}$ equivalent?

A1: No!

A2: Yes!

A4: One needs more information.