Let A be a nonsingular square matrix.
(a) Prove that the product of the singular values of A equals the absolute value of its determinant:
σ1 σ2 ... σn = |det A|.
(b) Does their sum equal the absolute value of the trace: σ1 + ... + σn = |tr A|?
(c) Show that if det A < 10-k, then its minimal singular value satisfies σn < 10-kn.
(d) True or false: A matrix whose determinant is very small is ill-conditioned.
(e) Construct an ill-conditioned matrix with det A = 1.