The problems in this paper have a wide range of difficulties.  They are specially designed to strengthen students’ concepts and to develop the problem solving skills.  Students are expected to drill the prob- lems with perseverance.

1. a and B are nonzero square matrices. If a and B commute, show that eA eB  = eA+B

2.  Given a function f(z), where

f(z) = │ 3      z(1)   1  │

Find the coefficient of z4  and z3  respectively without expansion.

(6 marks)

(6 marks)

3. Let a be a square matrix which satisfies a3 + a = a2 + Ⅰ .  Show that an + an −2  = a2 + Ⅰ , where n 』3. Find a103  if a =      1     .                                                         (10 marks)

4. Let a be an invertible matrix of order n 』  2.   Denote the adjoint of a by adj a and the determinant of a by 卜a卜.

(a)  Show that adj a is invertible and find (adj a)− 1 .                                                 (4 marks)

(b)  Show that 卜adj a卜 = 卜a卜n − 1 .                                                                                  (4 marks)

(c)  Show that adj (adj a) = 卜a卜n −2 a.                                                                        (4 marks)

(d) Find the conditions that adj (adj a) = a.                                                            (4 marks)

5.  Given a set of homogeneous linear equations:

            z1 + z2 + z3     =   0

(*)       α z1 + ó z2 + c z3     =   0

( α2 z1 + ó2 z2 + c2 z3     =   0

(a) Find the conditions of α , ó and c such that (*) has a trivial solution.                (8 marks)

(b) Find the conditions of α , ó and c such that (*) has infinitely many solutions. Obtain the

solutions.                                                                                                               (8 marks)

6.  Given a set of linear equations:

,    2 z1 + λ z2  | z3     =   1

(?) ．        λ z1  | z2 + z3     =   2

( 4 z1 + 5 z2  | 5 z3     =   |1

(a) Find λ such that (?) has an unique solution.

(b) Find λ such that (?) has infinitely many solutions. Obtain the solutions.

(c) Find λ such that (?) has no solution.

(6 marks)

(6 marks)

(6 marks)