In 1943, the document containing Red Army serviceman's service record was called "книжка красноармейца" (lit. "Red Army man's book"). It was introduced by the order №330 of People's Commissar of Defense of the Soviet Union.
According to the article 9 of that document, documents of soldiers leaving service were to be transferred to the unit's command for disposal. Soldiers were to receive "военный билет" at that point (the "military ID" from the OP's comment, but at that point it was more of a reservist's ID).
As Matrosov died, his book would be sent to the 91st rifleman brigade's command, where relevant information would be included in unit's irrecoverable casualty statement, and then the document would be disposed of as per order №330.
Thus, no, the document you are looking for isn't available, as it was destroyed back in 1943.