PHZ3113–Introduction to Theoretical PhysicsFall 2008Problem Set 11Oct. 15, 2008Due: Friday, Oct. 17, 2008Reading: Boas Ch. 31. The three Pauli matrices areσ1=·0 11 0¸; σ2=·0 −ii 0¸; σ3=·1 00 −1¸. (1)Show they have the properties σ2i= 1 for any i, and σiσj= i²ijkσk( i 6= j, k issummed, the i in front of the ²ijkand in σ2is√−1).2. Find the inverse matrices, using the formula with the cofactor matrix, andidentifying the cofactor matrix along the way:(a)A =·8 −23−412¸(2)(b)B =1 0 00 cos α −sin α0 sin α cos α(3)3. (a) Find the adjoint A†and transpose ATof the matrixA =1 0 01 + i 0 1 − i0 0 1(4)(b) Find a (any) 2 × 2 matrix which is self-adjoint A = A†.4.~v1= [3 0 4 − 1] ; A =2 −1 0 0−4 1 0 13 0 −3 12 2 0 0; ~v2=12−12. (5)Calculate A~v2and ~v1· A ·~v2.15. Solve the following system of equations using Cramer’s rule:3x + 3y + 3z = 0 (6)3x − 10y + 7z = 13 (7)x + 5y + 3z = −6