For the first problem you want all a such that x[1 a a]+ y[a 1 a]+ z[a a 1]= [1 1 1] forsomenumbers x, y, z. That gives the three equations x+ ay+az= 1, ax+y+az= 1, and ax+ay+ z= 1. For what values of a does that have a solution?

For the second problem you want to look at x[1 a a]+ y[a 1 a]+ z[a a 1]= [0 0 0]. An obvious solution is x=y= z= 0. For what values of a is that theonlysolution?

Are you saying you get all zeros in the last row of theaugmentedmatrix, for all a?