8.3. MULTILINEAR MAPS AND DETERMINANTS367Example 8.3.17.An element of Matn.Z/has an inverse in Matn.Q/if itsdeterminant is nonzero. It has an inverse in Matn.Z/if, and only if, itsdeterminant is˙1.Permanence of identititiesExample 8.3.18.For any be ann–by–nmatrix, let˛.A/denote the trans-pose of the matrix of cofactors ofA. I claim that(a)det.˛.A//Ddet.A/n1, and(b)˛.˛.A//Ddet.A/n2A.Both statements are easy to obtain under the additional assumption thatRis an integral domain and det.A/is nonzero. Start with the equationA˛.A/Ddet.A/E, and take determinants to get det.A/det.˛.A//Ddet.A/n. Assuming thatRis an integral domain and det.A/is nonzero,we can cancel det.A/to get the first assertion. Now we have˛.A/˛.˛.A//Ddet.˛.A//EDdet.A/n1E;as well as˛.A/ADdet.A/E. It follows that˛.A/ ˛.˛.A//det.A/n2AD0:Since det.A/is assumed to be nonzero,˛.A/is invertible in Matn.F /,whereFis the field of fractions ofR. Multiplying by the inverse of˛.A/gives the second assertion.
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