This will insert new Name fields that were not previously in the Matrix table.
INTO AdminAccounts (Name)
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;
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.