(a) State Brouwer's fixed-point theorem in 2 dimensions.

(b) State an equivalent theorem on retraction and explain (without detailed calculations) why it is equivalent.

(c) Suppose that $A$ is a $3 \times 3$ real matrix with strictly positive entries. By defining an appropriate function $f: \triangle \rightarrow \triangle$, where

$$\triangle=\left\{\mathbf{x} \in \mathbb{R}^{3}: x_{1}+x_{2}+x_{3}=1, x_{1}, x_{2}, x_{3} \geqslant 0\right\}$$

show that $A$ has a strictly positive eigenvalue.