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  "body": "\n## Proof\n\nWe compute $p_A(0)$ in two ways.\n\n1. Let $A$ be an $n \\times n$ matrix over $F$. The [[characteristic polynomial|039HF58KQ5]] is defined for $n \\ge 1$, so $n \\ge 1$. Assume that its characteristic polynomial is $p_A(t) = (t - \\lambda_1) \\cdots (t - \\lambda_n)$ with $\\lambda_i \\in F$.\n2. The [[evaluation homomorphism|0125QVNCFQ]] at $0$ preserves products. So $p_A(0) = (0 - \\lambda_1) \\cdots (0 - \\lambda_n) = (-1)^n \\lambda_1 \\cdots \\lambda_n$.\n3. By the rule for the [[determinant as a coefficient|0T0X8GTYYS]], $p_A(0) = (-1)^n \\det A$.\n4. By steps 2 and 3, $(-1)^n \\det A = (-1)^n \\lambda_1 \\cdots \\lambda_n$.\n5. Multiply both sides by $(-1)^n$, and use $(-1)^n (-1)^n = 1$. Then $\\det A = \\lambda_1 \\lambda_2 \\cdots \\lambda_n$.\n\n## Idea\n\nThe value of the characteristic polynomial at $0$ is $(-1)^n \\det A$. From the factored form, the same value is $(-1)^n$ times the product of the roots. The signs cancel.\n",
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        "locator": "Eigenvalues and characteristic polynomial",
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      "0T0X8GTYYS",
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