2 quick matrix questions.
A= (a b)
with none of a,b,c,d =0
If A is such that ad-bc=0, show the matrix equation AX+XA=0 has a solution X with X a non-zero 2x2 matrix that relies on 1 parameter only.
Ive written X as w,x,y,z and multiplied out then set the resulting equations in w,x,y,z to 0. It takes a few steps to get a relation between y and z which leads to relation between x,y and z then finally w,x,y and Z, so z=P is enough. My concern is that i dont use ad-bc=0 anywhere. what have i missed ? im guessing if A inverse exists we only get X=0??
2) a transformation im 3d space takes (a,b,c) to (x,y,z) where
(x)=(0 0 1)(a)
(y)=(1 0 0)(b)
(z)=(0 1 0)(c)
this transformation leaves distance between 2 points the same and leaves unaltered the points of the line x=y=z.
Assuming the transformation is a rotation about a line,find the angle of rotation.
the answer is 120 degrees but i cant find a decent way of explaining why. Im guessing the line of rotation is x=y=z and as lengths stay the same when i look at (1,0,0) which goes to (0,1,0) i get 2 equilateral triangles,one either side of the line.