Hatcher-Topology-of-Numbers · Problem 1.1.4

Continued Fractions

Mathematics — Algebraic Topology · matrix-column-reduction

Problem

(a) Reduce each of the matrices (73167)\begin{pmatrix} 7 & 3 \\ 16 & 7 \end{pmatrix} and (6714245)\begin{pmatrix} 67 & 14 \\ 24 & 5 \end{pmatrix} to either (1001)\begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} or (0110)\begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} by repeatedly subtracting one column from the other as in the proof of Proposition 1.1. (b) Use Proposition 1.1 to show that this can be done for any matrix (acbd)\begin{pmatrix} a & c \\ b & d \end{pmatrix} with nonnegative entries and determinant ±1\pm 1.

Related theory

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