u = $\begin{bmatrix} 1\\ 2 \end{bmatrix}$
v^{T}= $\begin{bmatrix} 1 & 1 \end{bmatrix}$
A= uv^{T}
A= $\begin{bmatrix} 1\\ 2 \end{bmatrix}$ $\begin{bmatrix} 1 & 1 \end{bmatrix}$
= $\begin{bmatrix} 1 &1 \\ 2 & 2 \end{bmatrix}$
A - λ I = 0
=> $\begin{bmatrix} 1 - \lambda &1 \\ 2 & 2- \lambda \end{bmatrix}$ = 0
=> (1- λ) (2- λ) - 2 =0
=> $\lambda ^{2} - 3\lambda = 0$
=> λ = 0, $3$
So maximum is $3$.