1. prove the following propositions. you may assume the following: . if a is an n × n upper or lower triangular matrix, then det (a) = a11 a22. . ann . det(ab) -det(a) det (b) for any n x n matrices a and b. det(a) det (a) for any n x n matrix a. in addition, you can assume any of the preceding propositions to the problem. for example, you may assume (a) and (b) to prove (c) (a) det (i) (b) det(ab) det (ba) (c) if a is non-singular, then det(a-1) 1/ det(a) (d) if a and b are similar, that is, there is a non-singular p with a = p-bp, then det (a) = det(b). (e) if q is an orthogonal matrix, that is, q-1-qt, then det(q) = ±1.