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The authors determine the number of ( n + m )× t matrices A ∗ of rank r + v , over a finite field GF ( q ), whose last m rows are those of a given matrix A of rank r + v over GF ( q ) and whose first n rows have a given rank u .

An expression is derived for the number of representations of one bilinear form by another (mod pa). From this, an explicit formula for the number of such representations is obtained in the case where both forms have square nonsingular matrices (mod p). A related bilinear analog of a lemma of Siegel on representations by quadratic forms (mod pa) is also proved.