  The paper studies algebraic strong shift equivalence of matrices over
$n$-variable polynomial rings over a principal ideal domain $D$($n\leq 2$). It
is proved that in the case $n=1$, every non-zero matrix over $D[x]$ has a full
rank factorization and every non-nilpotent matrix over $D[x]$ is algebraically
strong shift equivalent to a nonsingular matrix. In the case $n=2$, an example
of non-nilpotent matrix over $\mathbb{R}[x,y,z]=\mathbb{R}[x][y,z]$, which can
not be algebraically shift equivalent to a nonsingular matrix, is given.
