Here are two further references of proofs of the fixed point theorem that rely on evaluating determinants:

MR0117523 Dunford, Nelson ;  Schwartz, Jacob T.  Linear Operators. I. General Theory.
With the assistance of W. G. Bade and R. G. Bartle. Pure and Applied
 Mathematics, Vol. 7 Interscience Publishers, Inc., New York; Interscience Publishers, Ltd.,
 London  1958 xiv+858 pp.

MR0610487  Kannai, Yakar . An elementary proof of the no-retraction theorem.
 Amer. Math. Monthly  88  (1981),  no. 4, 264--268.