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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$$\\tr A = \\lambda_1 + \\lambda_2 + \\cdots + \\lambda_n.$$\n\n## Remarks\n\nThe $\\lambda_i$ are the [[eigenvalues|0BQAA2BK3Q]] of $A$, each repeated according to its [[algebraic multiplicity|0FY42N02DV]].\nThe hypothesis always holds over $\\C$.\nThe statement follows by comparing the coefficients of $t^{n-1}$, see [[the trace as a coefficient|0FRYPZ6D0X]].\nFor example, $\\begin{pmatrix} 2 \u0026 1 \\\\ 1 \u0026 2 \\end{pmatrix}$ has eigenvalues $1$ and $3$, and trace $4$.\n\n## Explanations\n\n### L1 human\n\nThe trace of a matrix is the sum of its eigenvalues, counted with multiplicity.\nFor a diagonal matrix this is clear, because the eigenvalues are the diagonal entries.\nThe statement says that it stays true for every complex matrix.\n",
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