In this paper we will present division of matrices and determinants. At division of determinants the determinants must have the same order n x n and the result will again be determinant of the same order n x n, and the final result is the same if we calculate determinants. While the division of the matrix as a result we will have a matrix of the same order m x n with the first respectively second matrix depending on whether it is right or left division. In order to apply this formula for division of matrices, they must have the same number of columns on right division or same number of rows on left division, the divisor matrix should be square matrix and its determinant should be different from zero.

A survey was carried out to determine activity concentration levels and associated hazard indices from the naturally occurring radionuclides Ra-226, Th-232, and K-40 in four different areas affected to varying degrees by TE-NORM activity by using of The High Purity Germanium Detector (HPGD). The average activity concentration levels due to radionuclides were measured for soil and fine dust samples collected from these areas. From the accumulated spectra, the average activity concentration were varied from 76.03 to 36.63, 43.79 and 10.63 Bq/Kg for Ra-226, 12.92 to 15.67, 15.77 and 12.94 Bq/Kg for Th-232, and 219.99 to 235.01, 205.03, and 204.06 Bq/Kg for K-40 for soil, and for fine dust were varied from 446.18 to 199.74, 45.84, and 6.8 Bq/Kg for Ra-226, 11.09 to 8.82, 10.58, and 12.94 Bq/Kg for Th-232, and 108.66 to 153.52, 169.39, and 204.06 Bq/Kg for K-40, of area (1), area (2), area (3), and area (4) respectively. The mean values of radium equivalent, absorbed dose rate (D_out), and Effective dose rate (D_eff) were 111.29, 77.14, 82.14, and 44.85 Bq/Kg, 52.24, 36.46, 38.58, and 21.46 (nGy/h), 0.06, 0.04, 0.05 and 0.03 (mSvyr^-1) for soil, and were 470.40, 224.18, 74.02, and 44.85 Bq/Kg, 217.55, 104.16, 34.81, and 21.46 (nGy/h), 0.26, 0.13, 0.04, and 0.03 (mSvy^-1) for fine dust of area (1), area (2), area (3), and area (4) respectively. Internal (H_in) and external (H_ex) hazard indices were calculated. In addition, the total effective dose from external, internal exposure and ingestion from uranium series (Ra-226) and thorium series (Ra-228) of different areas received by characteristic individuals were estimated.