Prove that each statement holds for 2 × 2 matrices.
(a) The determinant of a product is the product of the determinants det(ST) = det(S) ∙ det(T).
(b) If T is invertible then the determinant of the inverse is the inverse of the determinant det(T-1) = (det(T))-1.
Matrices T and T′ are similar if there is a nonsingular matrix P such that T′ = PTP-1. Show that similar 2 × 2 matrices have the same determinant.