Solution
= Solution
The $j$th column is
$$
c_j=v+aj\mathbf1,
\qquad
v=(1+b,2+b,3+b)^T.
$$
All three columns therefore lie in the two-dimensional span of $v$ and $\mathbf1$, so
$$
\boxed{\det A=0}.
$$
= Solution
The $j$th column is
$$
c_j=v+aj\mathbf1,
\qquad
v=(1+b,2+b,3+b)^T.
$$
All three columns therefore lie in the two-dimensional span of $v$ and $\mathbf1$, so
$$
\boxed{\det A=0}.
$$