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  "body": "\n## Definition\n\nLet $K$ be a commutative ring, let $n \\ge 1$, and let $A$ be an $n \\times n$ matrix over $K$.\nThe *characteristic polynomial* of $A$ is the polynomial\n$$p_A(t) = \\det(t I_n - A) \\in K[t],$$\nwhere $\\det$ is the [[determinant|00KDZRW4C0]] of a matrix with entries in the polynomial ring $K[t]$.\n\n## Remarks\n\nThe polynomial $p_A$ is monic of degree $n$.\nFor $A = \\begin{pmatrix} a \u0026 b \\\\ c \u0026 d \\end{pmatrix}$, $p_A(t) = t^2 - (a + d) t + (ad - bc)$.\nSome authors use $\\det(A - t I_n)$, which equals $(-1)^n p_A(t)$.\nOver a field, the roots of $p_A$ are the [[eigenvalues|0BQAA2BK3Q]] of $A$.\n\n## Explanations\n\n### L1 human\n\nThe number $\\lambda$ is an eigenvalue exactly when $A - \\lambda I$ is singular, that is, when $\\det(\\lambda I - A) = 0$.\nIf we replace $\\lambda$ by a variable $t$, then we get a polynomial.\nIts roots are the eigenvalues.\n",
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