In 1973, the UPI poll champion was crowned before the bowl games. Since Alabama was the UPI #1 after the regular season, Alabama claims 1973 as being a National Championship season, despite losing to Notre Dame in the Sugar Bowl. While Notre Dame was the consensus National Champion in 1973, Alabama still uses the UPI Championship as the basis for claiming a title in 1973.
However, I don't think that the Colley Matrix championship is recognized by the NCAA. According the NCAA website, the NCAA officially recognizes 13 seasons as being National Championship seasons for Notre Dame:
1988 - alone
1977- alone
1973 - shared with Alabama
1966 - shared with Michigan State
1964 - shared with Alabama and Arkansas
1949 - alone
1947 - alone
1946 - alone
1943 - alone
1930 - shared with Alabama
1929 - alone
1924 - alone
1919 - shared with Harvard, Illinois and Texas A&M