Generally, to apply the MUltiple SIgnal Classification (MUSIC) algorithm for the rapid imaging of small objects, complete elements of the multi-static response (MSR) matrix must be collected. However, in some real-world applications in microwave imaging, diagonal elements of the MSR matrix are unknown. Nevertheless, it is possible to obtain imaging results using a traditional approach but theoretical reason of the applicability has not been investigated yet. In this paper, we consider the application of MUSIC for a fast identification of small objects from collected MSR matrix in both transverse magnetic (TM) and transverse electric (TE) polarizations. In order to examine the applicability, fundamental limitation, and various properties of MUSIC, we establish mathematical structure of the three imaging functions and explore that the main factors of the imaging functions are Bessel function of order zero, one, and two. The established structures demonstrate why the existence and location of small objects can be retrieved via MUSIC without the diagonal elements of the MSR matrix. Results of numerical simulations with noise-corrupted synthetic data are also provided to support the identified structures.