Solutions To Simple Linear Algebra Problems (2)

 Problems:

1) Recall t^A, found in Ex. 1 and let B =

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

a) Find AB and thence: t^(AB)

b) Verify that: t^AB = t^B t^A

Solutions:

A =

(2...1)
(3...1)

t^A =

(2.....3)

(1.....1)

And let B =

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

Then AB = A  X B =

(a11 a12) (b11 b12)
(a21 a22) (b21 b22)

= [{(a11b11) + (a12b21)} --{(a11 b12)+ (a12 b22)} ]
[(a21b11) + (a22b21) } --{((a21 b12) + (a22 b22)}]

AB =

Then the transpose:

t^(AB) =  

(-1....-2)
(2.....3)

t^B  = 

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

t^A =  

(2.....3)

(1.....1)

Then:  t^B t^A  =

So that: 

t^AB = t^B t^A

2) Find the trace of: R3(Θ) =

(cos(Θ)..........sin(Θ)..........0)
(-sin (Θ)......cos(Θ)...........0)
(0 ..................0..................1)

Solution:

Tr R3(Θ) =  cos(Θ)  +  cos(Θ)   + 1  = 1 +  2 cos(Θ)