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  "body": "\n## Statement\n\nLet $F$ be a field and let $A$ be an $n \\times n$ matrix over $F$.\nAssume that the characteristic polynomial of $A$ splits over $F$ as $p_A(t) = (t - \\lambda_1) \\cdots (t - \\lambda_n)$.\nThen\n$$\\det A = \\lambda_1 \\lambda_2 \\cdots \\lambda_n.$$\n\n## Remarks\n\nThe $\\lambda_i$ are the [[eigenvalues|0BQAA2BK3Q]] of $A$, each repeated according to its [[algebraic multiplicity|0FY42N02DV]].\nThe statement follows by comparing the constant coefficients, see [[the determinant as a coefficient|0T0X8GTYYS]].\nFor example, $\\begin{pmatrix} 2 \u0026 1 \\\\ 1 \u0026 2 \\end{pmatrix}$ has eigenvalues $1$ and $3$, and determinant $3$.\n\n## Explanations\n\n### L1 human\n\nThe determinant of a matrix is the product of its eigenvalues, counted with multiplicity.\nThis agrees with the volume picture: in a basis of eigenvectors, the map stretches each direction by its eigenvalue.\nThe volume then changes by the product of these factors.\n",
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