Take coefficients of the variable from a Matrix

Total[D[A, {{a, b, c, d, e}}], {2}]

{{2, 3, 4, 0, 0}, {0, 0, 0, 4, 0}, {1, 0, 0, 0, 1}}

Also

Total[Transpose[Coefficient[A, #] & /@ {a, b, c, d, e}], {3}]

{{2, 3, 4, 0, 0}, {0, 0, 0, 4, 0}, {1, 0, 0, 0, 1}}

and

Transpose@Total[Coefficient[A, #] & /@ {a, b, c, d, e}, {3}]

{{2, 3, 4, 0, 0}, {0, 0, 0, 4, 0}, {1, 0, 0, 0, 1}}

Another way.

A = {{2 a, 3 b, 4 c}, {4 d}, {a, e}};
(* Variables[A]=={a,b,c,d,e} *)
A /. Thread[{a, b, c, d, e} -> IdentityMatrix[5]]
% // Map[Total]

{{{2, 0, 0, 0, 0}, {0, 3, 0, 0, 0}, {0, 0, 4, 0, 0}}, {{0, 0, 0, 4, 0}}, {{1, 0, 0, 0, 0}, {0, 0, 0, 0, 1}}}

{{2, 3, 4, 0, 0}, {0, 0, 0, 4, 0}, {1, 0, 0, 0, 1}}

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