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  "body": "\n## Statement\n\nLet $K$ be a commutative ring, let $n \\ge 1$, and let $A$ be an $n \\times n$ matrix over $K$.\nThen the coefficient of $t^{n-1}$ in the [[characteristic polynomial|0VSSPAFM2C]] $p_A(t) = \\det(t I_n - A)$ is $-\\tr A$, where $\\tr$ is the [[trace|0XC1RDBY7A]].\nSo $p_A(t) = t^n - (\\tr A)\\, t^{n-1} + \\cdots$.\n\n## Remarks\n\nIn the Leibniz expansion of $\\det(t I_n - A)$, only the product of the diagonal entries $(t - a_{11}) \\cdots (t - a_{nn})$ contributes to $t^{n-1}$.\nFor $n = 2$, $p_A(t) = t^2 - (\\tr A) t + \\det A$.\nTogether with [[the constant coefficient|0T0X8GTYYS]], it shows that $\\tr$ and $\\det$ are invariants of similarity.\n\n## Explanations\n\n### L1 human\n\nThe characteristic polynomial contains the trace as its second coefficient, with a minus sign.\nFor a $2 \\times 2$ matrix, the full polynomial is $t^2 - (\\text{trace})\\,t + \\text{determinant}$.\nSo the trace and the determinant are part of the information of the characteristic polynomial.\n",
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