l 2 2 7 1. (a) Find the determinant of the matrix 0 3 0 0 2 . l l 0 l sin(9) 3 cos(9) (b) Determine all values of 6 for which the matrix 0 2 O is invertible. cos(6) 3 sin(9) 2. Let A, B, C, and D be (n x n)-matrices with A invertible. (a) Show that dd\": 3]] =det(A)det(D). Hint: ['3 g] = ['3 (II] [II] AI; C] (b) Find matrices X and Y which produce the block LU factorization A C _ I 0 A C B D _ X I 0 Y ' Sh thtdt AC thdtD BA'1C (c) owaeBDe()e( ). (d) IfAB 2 BA then prove that det[ [I]: 3]] = det(AD BC). (e) IfAB 7E BA, then give an example such that det [ [B S] ] 7E det(AD BC). 3. (3) Compute the determinants of the following matrices: [i' 3] (b) Use the results in part (a) to guess the determinant of the (n your guess by using properties of determinants and induction. 013--- rIUJ O [ rIUJN MONO 00 00 00 32 13 X OOHW OHUJN 0 2 3 1 OJNOO n)-matrix below. Conrm