For what values of λ does Rank(A) = 1? Rank(A)= 2? Rank(A) = 3? $$ A = \begin{pmatrix} λ & 1 & 1 \\ 1 & λ & 1 \\ 1 & 1 & λ \end{pmatrix} $$
My attempt: 
For Rank(A) = 1, the rows of matrix A must be scalar multiples of each other. Thus, the only value that works in this case would be λ = 1. 
For Rank(A) = 2, the determinant of A must equal zero in order to find the values of λ. 
The determinant is λ^3 -3λ +2 = 0 which gives values of λ = -2 and λ = 1. Since we know that λ = 1 makes Rank(A) = 1, we can disregard λ = 1 in this case. Thus λ = -2 would make Rank(A) = 2. 
For Rank(A) = 3, the determinant of A must not equal zero. Since we found values of λ for which det(A) = 0, the values of λ for which Rank(A) = 3 should be numbers that do not equal 1 or -2. 
Am I missing any λ values for each case? 

Looks good to me. Note that this is a special case of a rank-1 update of a matrix.
