This paper presents an optimal relay amplifying matrix based on various schemes, such as zero-forcing (ZF) and minimum mean square error (MMSE), for a cooperative amplifyand-forward (AF) wireless relay network consisting of a one source-one-destination pair with a single antenna and one relay with N antennas. The total relay transmission power constraint is employed. Even though different schemes and methods are applied to determine the optimal relay amplifying matrix, the identical one is derived. By adopting the derived optimal relay
amplifying matrix, various analytical and numerical discussions, such as achievable rate (AR), are presented.