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  "body": "\n## Summary\n\nA determinantal ideal is the ideal generated by all minors of a fixed size of a matrix with entries in a commutative ring.\nTake the generic matrix, whose entries are independent variables.\nThen the zero set of the ideal is the determinantal variety of matrices of rank less than the size of the minors.\nThese ideals are among the best understood classes of ideals.\nFor the generic matrix they are prime, and their quotient rings are Cohen–Macaulay with a known height.\nThey appear in the study of resolutions, of rank conditions, and of secant varieties.\n\n## Explanations\n\n### L1 human\n\nA matrix has rank less than $t$ exactly when all of its $t \\times t$ minors vanish.\nSo the condition \"rank less than $t$\" is a system of polynomial equations in the entries.\nThe determinantal ideal is the ideal generated by those minors.\nFor a $2 \\times 2$ matrix of variables $\\begin{pmatrix} a \u0026 b \\\\ c \u0026 d \\end{pmatrix}$ and $t = 2$, it is the principal ideal $(ad - bc)$, and its zero set is the set of singular $2 \\times 2$ matrices.\n",
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