| Summary: | The determinant of a Mueller matrix <b>M</b> plays an important role in both polarization algebra and the interpretation of polarimetric measurements. While certain physical quantities encoded in <b>M</b> admit a direct interpretation, the understanding of the physical and geometric significance of the determinant of <b>M</b> (<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>det</mi><mstyle mathvariant="bold" mathsize="normal"><mi>M</mi></mstyle></mrow></semantics></math></inline-formula>) requires a specific analysis, performed in this work by using the normal form of <b>M</b>, as well as the indices of polarimetric purity (IPP) of the canonical depolarizer associated with <b>M</b>. We derive an expression for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>det</mi><mstyle mathvariant="bold" mathsize="normal"><mi>M</mi></mstyle></mrow></semantics></math></inline-formula> in terms of the diattenuation, polarizance and a parameter proportional to the volume of the intrinsic ellipsoid of <b>M</b>. We likewise establish a relation existing between the determinant of <b>M</b> and the rank of the covariance matrix <b>H</b> associated with <b>M</b>, and determine the lower and upper bounds of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>det</mi><mstyle mathvariant="bold" mathsize="normal"><mi>M</mi></mstyle></mrow></semantics></math></inline-formula> for the two types of Mueller matrices by taking advantage of their geometric representation in the IPP space.
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