An interesting class of Banach algebras, the G-completely symmetric Banach algebras [see the author, Kexue Tongbao 24, 385-388 (Chinese) (1979; Zbl 0437.46037)] was exhibited in this paper so as to show that an elliptic pseudodifferential operator matrix of order N is a Fredholm operator on \(L^ p_ n(R^ n) (p>1)\) if and only if the determinant of the associated symbol matrix vanishes nowhere. For the special cases, \(p=2\) or \(N=1\), the result is well known.