= Solution
A direct <determinant> computation gives
$$
\det(tI-D)=t^2(t-4),\qquad \operatorname{rank}D=2.
$$
The zero <eigenvalue> has algebraic multiplicity two but a one-dimensional <eigenspace>, so it gives one <Jordan block> of size two. The other block is $J_1(4)$. For an explicit <Jordan basis>, take
$$
v=(1,-3,1)^T,\quad u=(4,-7,0)^T,\quad w=(1,1,1)^T.
$$
These are linearly independent and satisfy $Dv=0$, $Du=v$, $Dw=4w$. Thus
$$
\boxed{{}_AM\cong A/(X^2)\oplus A/(X-4).}
$$
The first summand has the chain $u\mapsto v\mapsto0$ and is an <indecomposable module>; the other is simple.
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