  We consider the limiting location and limiting distribution of the largest
eigenvalue in real symmetric ($\beta$ = 1), Hermitian ($\beta$ = 2), and
Hermitian self-dual ($\beta$ = 4) random matrix models with rank 1 external
source. They are analyzed in a uniform way by a contour integral representation
of the joint probability density function of eigenvalues. Assuming the one-band
condition and certain regularities of the potential function, we obtain the
limiting location of the largest eigenvalue when the nonzero eigenvalue of the
external source matrix is not the critical value, and further obtain the
limiting distribution of the largest eigenvalue when the nonzero eigenvalue of
the external source matrix is greater than the critical value. When the nonzero
eigenvalue of the external source matrix is less than or equal to the critical
value, the limiting distribution of the largest eigenvalue will be analyzed in
a subsequent paper. In this paper we also give a definition of the external
source model for all $\beta$ > 0.
