Solving matrix algebra

Question

Question:

Matrix algebra

1.Given the matrices

A : = (0 1 0) ,     B: = (1 0 0) ,    C:= (1  0  0)

         (1 0 1)              (0 1 0)             (0  0  0)

         (0 1 0)              (0 0 1)             (0  0 -1)

Show that A and B commute, B and C commute but A and C do not.

2.Show that the matrix
                          A: =   (1 4 0)
                                   (2 5 0)
                                   (3 6 0)

 Is not invertible

3.Find the inverse of the matrix

                         A: = (1 2 3)
                                 (2 5 3)
                                 (1 0 8)

And confirm your result by direct calculation.

4.Let A,B be aritrary matrices whose product AB exists.

a. Show that AA^† and A^†A exist and Hermitian (^† represent complex conjugate)

b. Show that the product (B^†A^†) exists and equal to (AB)^†

c. If both A and B are Hermitian, show that AB+BA is Hermitian as well

d. If A and B are both Hermitian, show that (i(AB-BA) is Hermitian as well.

5. The Pauli spin matrices

σ1 : = ( 0 1) ,    σ2 : =( 0 -i)    and  σ3 : = (1 0)
          ( 1 0)                (i   0)                      (0 -1)

Play an important role in quantum mechanics.

a. show that each of the Pauli spin matrices is both Hermitian and unitary. Calculate teh inverse of each.

b. Show that the product of two Pauli matrices is

                       σ_iσ_j = δ_ij I + i Σ_k ∈ijkσk,

where δ_ij and ∈_ijk are the Kronecker and Levi-Civita symbols, respectively.

c. Calculate the commutator [σ_i,σ_j ] of two Pauli matrices

6.Let A, B and C be square matrices of the same dimension. Show that tr AB = tr BA and tr ABC = tr BCA = tr CAB.

            Is tr ACB = tr ABC?