Inverse spectrum problem - showing the existence of a 2x2 doubly stochastic matrix,

Question

I am working through a couple of problems in Henryk Minc's book, Nonnegative Matrices, as a warm-up to understanding the inverse spectrum problem.

This is Exercise 18 of Chapter VII of his book:

Show that for any real number $\lambda_2$ such that $|\lambda_2|\le1$, there exists a $2x2$ doubly stochastic matrix with eigenvalue $\lambda_2$.

Any hints or suggestions are welcome.

I started with this doubly stochastic matrix and was trying to see whether it could offer some insight:

$$ \large \begin{bmatrix} {1-\lambda_2} & \lambda_2\\ \lambda_2 & {1-\lambda_2} \\ \end{bmatrix} $$

Thanks,