Try to solve |T- $\lambda$I | = 0. In first step perform c1 <- c1 + c2 +c3+c4+c5 and after that all elements of first column would be 1+2+3+4+5-$\lambda$ = 15-$\lambda$. Now we take 15-$\lambda$ factor out of the determinant. Therefore one real eigen value would be 15.