There are many properties of determinants detailed in this section. You have the tools in Sage to explore all of them. We will illustrate with just two. One is not particularly deep, but we personally find the interplay between matrix multiplication and the determinant nothing short of amazing, so we will not pass up the chance to verify Theorem DRMM. Add some checks of other properties yourself.
We copy the sixth row of the matrix C to the fourth row, so by Theorem DERC, the determinant is zero.
Now, a demonstration of Theorem DRMM.
Earlier, in Sage NME1 we used the random_matrix() constructor with the algorithm='unimodular' keyword to create a nonsingular matrix. We can now reveal that in this context, “unimodular” means “with determinant 1.” So such a matrix will always be nonsingular by Theorem SMZD. But more can be said. It is not obvious at all, and Solution PDM.M20 has just a partial explanation, but the inverse of a unimodular matrix with all integer entries will have an inverse with all integer entries.
With a fraction-free inverse many “textbook” exercises can be constructed through the use of unimodular matrices. Experiment for yourself below. An upper_bound keyword can be used to control the size of the entries of the matrix, though this will have little control over the inverse.