$$C_{n\times n}={\begin{bmatrix} 1 & x_{12} & \dots & x_{1n} \\ {x_{12}}^{-1} & 1 & \dots & x_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ {x_{1n}}^{-1} & {x_{2n}}^{-1} & \dots & 1 \end{bmatrix}}$$
Criteria
$$x_{ab} = \frac{1}{x_{ba}}, \qquad x_{ab}>0, \qquad x_{aa} = 1$$
Alternatively
$$C \cdot C^{T} = \underline1$$
I know that this matrix models currency exchange (without commission or fluctuation). So I'm guessing it's called a currency matrix or a trade matrix. I'm just after a name so I can search its properties. I couldn't spot it in the Matrix Reference Manual.
I'm curious about some of its algebraic properties, for example, it has only 1 non-zero eigenvalue. What does this eigenvalue signify?
$$\text{Eigenvalues of} \begin{bmatrix} 1 & 6 & 30 & 210 \\ 1/6 & 1 & 5 & 35 \\ 1/30 & 1/5 & 1 & 7 \\ 1/210 & 1/35 & 1/7 & 1 \end{bmatrix} = 0,0,0,4$$
Any help, ideas, advice greatly welcome.
$C$ is a positive reciprocal matrix
to be an exchange matrix the additional condition that $a_{ij} \times a_{jk} = a_{ik}$ is required
$C$ is then a Positive Saaty-consistent Reciprocal matrix (PS-cR)
the eigenvalues of an $n\times n$ PS-cR matrix are $n,0,..,0$
Thank you @Algebraic Pavel for the term reciprocal.