1. The problem statement, all variables and given/known data If A2 is a zero matrix, find all symmetric 2x2 nilpotent matrices. 2. Relevant equations 3. The attempt at a solution So if A2 is nilpotent, then [a,b;c,d]*[a,b;c,d] is equal to [0,0;0,0]. Since A is symmetric, b=c. Multiplying the two matrices, I get [ aa + bb, ab + bd; ba +db, bb + dd] = [0,0;0,0] each element in the matrix must equal zero, so aa + bb = 0 ab + bd = 0 ba + bd = 0 bb + dd = 0 with the first equation, a2 must equal negative b2, so there is no solution. Is there no 2x2 symmetric nilpotent matrices, or did I mess up somewhere?