INSERT INTO without duplicates

This will insert new Name fields that were not previously in the Matrix table.

INSERT IGNORE
INTO AdminAccounts (Name) 
SELECT Name
FROM Matrix;

If you do not trust INSERT IGNORE, there is an alternative where you can manifest the new Name values before inserting them:

CREATE TABLE NewName SELECT Name FROM Matrix WHERE 1=2;
INSERT INTO NewName
    SELECT Name FROM AdminAccounts A
    LEFT JOIN Matrix B USING (Name)
    WHERE B.Name IS NULL;
INSERT INTO Matrix (Name) SELECT Name FROM NewNames;

The table NewName collects only those tables in AdminAccounts that are not in Matrix at present. This can give you a chance to look over the new Names. Afterwards, you can INSERT everything in NewName into Matrix.

INSERT INTO AdminAccounts 
    (Name)
SELECT t1.name
  FROM Matrix t1
 WHERE NOT EXISTS(SELECT id
                    FROM AdminAccounts t2
                   WHERE t2.Name = t1.Name)