The concept of “Eigenvalues” of a matrix is an important concept in Linear Algebra with various applications. Out of the many applications of “Eigenvalues” of a matrix, an important application is that in the field of “Numerical Analysis”. This chapter discusses the use of “Eigenvalues” in determining the convergence of “iterative methods for solution of system of linear equations”. The chapter focusses on the discussion of two important iterative methods for obtaining solution of “system of linear equations”, namely “the Gauss-Jacobi Method” and “the Gauss-Seidel Method”. Numerous results for determining convergence of the given iterative method for linear systems have been presented. The discussed results exhibit the use of the concept of the “spectral radius” of the iteration matrix associated with the given iterative method for establishing the method‘s convergence.