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.