Postgres 13

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CREATE TABLE tbl (arr int[]);
INSERT INTO tbl VALUES
  ('{1,2,3}')
, ('{9,12,13}')
, ('{1,1,1, 33}')  -- 4h element
, ('{NULL,NULL}')  -- NULL elements
, (NULL)           -- NULL array
, ('{}')           -- empty array
;
6 rows affected

-- Postgres 9.4+
SELECT ARRAY (
   SELECT sum(elem)
   FROM  tbl t
       , unnest(t.arr) WITH ORDINALITY x(elem, rn)
   GROUP BY rn
   ORDER BY rn
   );


array


{11,15,17,33}


-- Postgres 9.3+
SELECT ARRAY (
   SELECT sum(arr[rn])
   FROM   tbl t
        , generate_subscripts(t.arr, 1) AS rn
   GROUP  BY rn
   ORDER  BY rn
   );


array


{11,15,17,33}


-- Postgres 9.1
SELECT ARRAY (
   SELECT sum(arr[rn])
   FROM  (
      SELECT arr, generate_subscripts(arr, 1) AS rn
      FROM   tbl t
      ) sub
   GROUP  BY rn
   ORDER  BY rn
   );


array


{11,15,17,33}