Solve Matrix A: Determining Trace & R(LA)

[steinmasta]

Homework Statement

Suppose A is a real symmetric 3
× 3 matrix such that
• trace(A) = 0
• R(LA) = span {(1, 1, 1) and (1, 0, -1)} (sorry for formatting issues - these are both column vectors)
where La is the left multiplication transformation

• A * (1, 1, 1) = (2, 1, 0) again, these are column vectors

Find A. Explain your answer.

Attempts at solution:
Because the matrix is symmetric, entries a12 = a21, a13=a31 and a23=a32, so there are 6 unknowns.

I need 6 equations then. The trace being zero implies a11+a22+a33 = 0, so that's one equation. Three more equations come from the last condition, multiplying A * the column vector (1, 1, 1). I need two more equations, which I think come from the condition regarding R(La), but I can't figure them out. Any help would be much appreciated.

 

[mutton]

What does R denote? Does this mean that for all x, Ax = b(1, 1, 1) + c(1, 0, -1) for some real b, c?