Linear Systems: Consistency and Compatibility

Let $A$ represent an $m \times n$ matrix, $C$ an $n \times q$ matrix, and $B$ a $q \times p$ matrix. Show that if $\mathrm{rank}(AC) = \mathrm{rank}(C)$, then

and that if $\mathrm{rank}(CB) = \mathrm{rank}(C)$, then

Let $A$ and $B$ represent $m \times n$ matrices. (1) Show that if $C$ is an $r \times q$ matrix and $D$ a $q \times m$ matrix such that $\mathrm{rank}(CD) = \mathrm{rank}(D)$, then $CDA = CDB$ implies $DA = DB$.

(2) Similarly, show that if $C$ is an $n \times q$ matrix and $D$ a $q \times p$ matrix such that $\mathrm{rank}(CD) = \mathrm{rank}(C)$, then $ACD = BCD$ implies $AC = BC$.