This question has a lot data but I don't really know how to connect it all together 1. The problem statement, all variables and given/known data Let A be a 5x5 matrix over R 3 eigenvectors of A are: [itex]u_1=(1,0,0,1,1)[/itex] [itex]u_2=(1,1,0,0,1)[/itex] [itex]u_3=(-1,0,1,0,0)[/itex] also: [itex]\rho (2I-A)>\rho (3I-A)[/itex] and: [itex]A(1,2,2,1,3)^t=(0,4,6,2,6)^t[/itex] prove that A is diagonalizable and find a diagonal matrix similiar to it. 2. Relevant equations 3. The attempt at a solution What I can make of this is: [itex](1,2,2,1,3)^t=u_1 + 2u_2 + 2u_3[/itex] (then maybe I can say that P has the eigenvectors as columns and [itex]AP(1,2,2,0,0)^t=(0,4,6,2,6)^t[/itex] but then what?) and because [itex]\rho (2I-A)>\rho (3I-A)[/itex] [itex]\rho (2I-A)\leq 5[/itex] it means that [itex]5>\rho (3I-A)[/itex] and so [itex]det(3I-A)=0[/itex] and 3 is an eigenvalue of A. I would like some hints/suggestions on what to do. Thanks!