Let A =

[ 3 0 1 ]

[-4 1 -2 ]

[ -4 0 -1 ] which is a 3x3 matrix. and let I be a 3x3 identity matrix.

Find a basis for the intersection of R(A-I) and N(A-I), where R(A-I) is the range of A-I and N(A-I) is the nullspace of A-I.

the solution says this basis is [1 -2 -2]^T (this T means transpose).

and the procedure is the following.

1. Find a basis {x1,x2,..,} for R(A-I)

2. Let X be the matrix that consists of the vectors in this basis for R(A-I).

3. Find a basis {v1,v2,....,}for N((A-I)X)

4. B={Xv1, Xv2,...,..} is the basis of the intersection.

i ve been trying to do this problem by following the procedure but i cant get the solution and i dont know what i am doing wrong. please help me.can please anyone explain and show me how to find this basis?