Using similar tools and det (I + abT ) = 1 + bT a, find the derivative of the determinant of a matrix with respect to its elements. (It will be a matrix.) By rescaling the rows and/or columns of the...

Using similar tools and det (I + abT ) = 1 + bT a, find the derivative of the determinant of a matrix with respect to its elements. (It will be a matrix.)

By rescaling the rows and/or columns of the inverse of the Hilbert matrix used in Example 3.5 and chex35, can you improve the condition number?