Let S = [ x x x x ; a x x x ; 0 b x x ; 0 0 c x] Find the rank of S-K*identity. Attempt I basically did straight forward row reduction and got [ (x-k) x x x ; 0 ((x-k)^2-ax)/(x-k) x-ax/(1-k) a- ax/(1-k); 0 0 (x-k)-b[x(x-k)-ax]/ [(x-k)^2-ax] (x-k)-b[x(x-k)-ax]/[(x-k)^2-ax]; 0 0 0 [(x-k)^2-ax(x-k)-c]/[(x-k)^2 -ax] ] It is given that the rank is 3. I am not sure how to prove this. Shouldn't I have gotten a zero row when row reducing? Are there any alternatives to find the rank of a matrix like finding the upper bound of the rank is 3 and the lower bound for the rank is 3?