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In this paper, the Rayleigh's quotient and the inverse vector iteration method are presented. The latter approach helps to obtain the natural frequencies and mode shapes of a structure. Inverse vector iteration method with shifting enables to determine the higher modes. Some basic theorems of linear algebra are presented and extended to study the free vibration of structures. The variation theorems are presented for predicting the eigenvalues and eigenvectors of the modified structures. These theorems reduce the number of cycles of the iterations used for calculating the eigenvalues and eigenvectors of the modified structures. Finally, an example is solved to show the ability of the present approach.