I am trying to determine whether or not the set of all n x n diagonal matrices under matrix multiplication is a group. I can show that the set is closed and associative under the operation, but I am confused on how to show that every element of the set has an inverse in the set. I understand that since the determinant of any element of the set is nonzero each element has an inverse, but why must that inverse belong to the set?
you seem to have neglected the "the determinant of the matrix is non-zero" from the beginning of the question. If you are considering the collection of diagonal matrices, then it is not a group. But the collection of invertible diagonal (i.e., those diagonal matrices with non-zero determinant), then it is indeed a group. To see that this collection is closed under inverses, take an arbitrary diagonal matrix $diag(d_1,\ldots, d_n)$ with non-zero determinant. What does that tell you about each $d_i$? Can you now explicitly describe the inverse matrix and see that it too is diagonal?