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  "body": "\n## Definition\n\nLet $K$ be a commutative ring and let $n \\ge 1$.\nThere is exactly one map $\\det \\colon M_n(K) \\to K$ with these properties:\n\n1. $\\det A$ is a multilinear function of the $n$ rows of $A$.\n2. $\\det A = 0$ when two rows of $A$ are equal.\n3. $\\det I_n = 1$, for the [[identity matrix|06DW9BAWB1]] $I_n$.\n\nThis map is the *determinant*.\n\n## Remarks\n\nIn other words, $\\det$ is the unique [[alternating multilinear map|0K6CH0MT0F]] of the rows with value $1$ on $I_n$.\nThe [[Leibniz formula|016PE4VZ8Q]] gives $\\det A$ as a sum over permutations.\nFor $n = 2$, $\\det \\begin{pmatrix} a \u0026 b \\\\ c \u0026 d \\end{pmatrix} = ad - bc$.\nThe same characterization holds with columns in place of rows, because $\\det A^T = \\det A$.\n\n## Explanations\n\n### L1 human\n\nThe determinant gives one number for each square matrix.\nOver the real numbers, its absolute value is the factor by which the matrix scales volume, and its sign tells if the matrix flips orientation.\nA real square matrix is invertible exactly when its determinant is not zero.\n",
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