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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 constant coefficient of the [[characteristic polynomial|0VSSPAFM2C]] $p_A(t) = \\det(t I_n - A)$ is\n$$p_A(0) = (-1)^n \\det A,$$\nwhere $\\det$ is the [[determinant|00KDZRW4C0]].\n\n## Remarks\n\nThe proof sets $t = 0$: $p_A(0) = \\det(-A) = (-1)^n \\det A$.\nSo, over a field, $0$ is an eigenvalue of $A$ if and only if $\\det A = 0$.\n\n## Explanations\n\n### L1 human\n\nThe constant term of the characteristic polynomial is the determinant, up to a sign.\nThis follows when we put $t = 0$.\nSo a matrix with determinant zero has the eigenvalue $0$.\n",
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