035: Molnar's Problem

Proposed by Alan Frisch, Chris Jefferson, Ian Miguel

Molnar originally posed the following problem to construct a $k \times k$ matrix

$$\begin{pmatrix} a_{11} & \ldots & a_{1k}\\ \vdots & \ldots & \vdots\\ a_{k1} & \ldots & a_{kk} \end{pmatrix}$$

such that:

$$\det \begin{pmatrix} a_{11} & \ldots & a_{1k}\\ \vdots & \ldots & \vdots\\ a_{k1} & \ldots & a_{kk} \end{pmatrix} = 1, \ \det \begin{pmatrix} a_{11}^2 & \ldots & a_{1k}^2 \\ \vdots & \ldots & \vdots \\ a_{k1}^2 & \ldots & a_{kk}^2 \end{pmatrix} = \pm 1$$

where the $a_{ii}$ are integers not equal to plus or minus 1, and $\det$ denotes the determinant of a matrix. The solutions to this problem are significant in classifying certain types of topological spaces. Guy discusses a variant where 0 entries are also disallowed and the sign of both determinants must be positive.