Let \( \mathrm{A} \) be an idempotent matrix and \( (\mathrm{I}+\ma...

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Let \( \mathrm{A} \) be an idempotent matrix and \( (\mathrm{I}+\mathrm{A})^{100}=\mathrm{I}+\left(2^{20 \mathrm{k}}-1\right) \mathrm{A} \), then \( k=\ldots \)
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