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  "body": "\n## Definition\n\nLet $n \\ge 1$ and $N \\ge 1$, and let $p_0, \\ldots, p_n$ be points of $\\R^N$.\nWrite $d_{ij} = \\norm{p_i - p_j}$ for their [[Euclidean distances|0C95TG69SG]].\nThe *Cayley–Menger determinant* of the points is the [[determinant|0C3CFHDN6R]] of the $(n + 2) \\times (n + 2)$ matrix\n$$\\operatorname{CM}(p_0, \\ldots, p_n) = \\det \\begin{pmatrix} 0 \u0026 1 \u0026 1 \u0026 \\cdots \u0026 1 \\\\ 1 \u0026 0 \u0026 d_{01}^2 \u0026 \\cdots \u0026 d_{0n}^2 \\\\ 1 \u0026 d_{10}^2 \u0026 0 \u0026 \\cdots \u0026 d_{1n}^2 \\\\ \\vdots \u0026 \\vdots \u0026 \\vdots \u0026 \\ddots \u0026 \\vdots \\\\ 1 \u0026 d_{n0}^2 \u0026 d_{n1}^2 \u0026 \\cdots \u0026 0 \\end{pmatrix}.$$\n\n## Remarks\n\nThe $n$-dimensional volume $V_n$ of the simplex with vertices $p_0, \\ldots, p_n$ satisfies\n$$V_n^2 = \\frac{(-1)^{n+1}}{2^n (n!)^2} \\operatorname{CM}(p_0, \\ldots, p_n).$$\n\nExample: for $n = 1$ and two points at distance $d$, the matrix is $\\begin{pmatrix} 0 \u0026 1 \u0026 1 \\\\ 1 \u0026 0 \u0026 d^2 \\\\ 1 \u0026 d^2 \u0026 0 \\end{pmatrix}$.\nIts determinant is $2d^2$, and the formula gives $V_1^2 = \\frac{1}{2} \\cdot 2d^2 = d^2$, the squared length.\n\nExample: for $n = 2$ and an equilateral triangle of side $1$, the matrix is $J - I$ with $J$ the $4 \\times 4$ all-ones matrix.\nIts eigenvalues are $3, -1, -1, -1$, so the determinant is $-3$.\nThe formula gives $V_2^2 = \\frac{-1}{16} \\cdot (-3) = \\frac{3}{16}$, the square of the area $\\frac{\\sqrt{3}}{4}$.\n\nDegenerate case: if the points are affinely dependent, such as three points on a line, the determinant is $0$.\nThe border of ones is essential: the determinant of the matrix of squared distances alone does not give the volume.\n",
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