A[λ_]:= {
{1 + 2λ, -λ, 0, -λ, 0, 0, 0, 0, 0, 0, 0, 0},
{- λ, 1 + 3λ, -λ, 0, -λ, 0, 0, 0, 0, 0, 0, 0},
{0, - λ, 1 + 2λ, 0, 0, -λ, 0, 0, 0, 0, 0, 0},
{-λ, 0, 0, 1 + 3λ, -λ, 0, -λ, 0, 0, 0, 0, 0},
{0, -λ, 0, -λ, 1 + 4λ, -λ, 0, -λ, 0, 0, 0, 0},
{0, 0, -λ, 0, -λ, 1 + 3λ, 0, 0, -λ, 0, 0, 0},
{0, 0, 0, -λ, 0, 0, 1 + 3λ, -λ, 0, -λ, 0, 0},
{0, 0, 0, 0, -λ, 0, -λ, 1 + 4λ, -λ, 0, -λ, 0},
{0, 0, 0, 0, 0, -λ, 0, -λ, 1 + 3λ, 0, 0, -λ},
{0, 0, 0, 0, 0, 0, -λ, 0, 0, 1 + 2λ, -λ, 0},
{0, 0, 0, 0, 0, 0, 0, -λ, 0, -λ, 1 + 3λ, -λ},
{0, 0, 0, 0, 0, 0, 0, 0, -λ, 0, -λ, 1 + 2λ}}

b = {1,4,1,2,-1,3,-4,-3, 2, 5, 2, 1}

A[λ]//MatrixForm

N[Inverse[A[2]].b]

Manipulate[ListPlot[{b,Inverse[A[λ]].b}, PlotRange->{-4,  5}], {λ, 0, 10}]