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