Solve Matrix equations with Cross Product: weird system of equations

I would like to find the values {Pfx, Pfy, Pfz} that satisfy the equation A X B = C . The code of everything is at the end, I want to ilustrate with images what i think of first:

A is this:

Matrix A

B is this: {0,0,0}

And C is this: {100,500,200}

The image of the complete code is this one:

Complete code

My variable here are {Pfx,Pfy,Pfz} and the {i,j,k} are the unit vectors. The way that the matrix is shown in the picture corresponds with a trick used to solve in papper this type of matrix.

The solution would give me the value of Pfx in the "x" coordinate (and this would be expressed by Pfx being multiplied with the i vector). And the same mechanism applies with Pfy related with j Vector ; and Pfz related with k.

The problem here comes with the fact that the I can`t find the values {Pfx, Pfy, Pfz} that satisfy the equation A X B = C. I am not sure if the problem lies in the "LinearSolve" comand, in the use of the CrossProduct or in the use of the versor {i, j, k} inside the matrix.

Any kind of help in this regard will be extreamly useful, thanks in advance!!


i := {1, 0, 0}  j := {0, 1, 0}  k := {0, 0, 1}  LinearSolve[({ {i, j, k},{1, 2, 3},{Pfx, Pfy, Pfz}})\[Cross]({{0},{0},{0}}) == ( {{100},{500},{200}} )]