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The lower bounds are best possible, and in several cases they are stronger than well-known bounds due to Ostrowski and other authors. If $A = I-E$ is an $n \\times n$ matrix and the elements of $E$ are bounded in absolute value by $\\varepsilon \\le 1/n$, then a lower bound of Ostrowski (1938) is $\\det(A) \\ge 1-n\\varepsilon$. 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