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(Solved): 1. What is the cardinality of the set of all 33 matrices with rational entries? Give a detail ...


1. What is the cardinality of the set of all \( 3 \times 3 \) matrices with rational entries? Give a detailed proof of your a

1. What is the cardinality of the set of all matrices with rational entries? Give a detailed proof of your answer


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Explanation :



To determine the cardinality of the set of all    matrices with rational entries, we first need to determine how many possible choices we have for each entry of the matrix. Since each entry can be any rational number, we can represent it as a fraction of the form p/q, where p and q are integers with   


For the first entry in the first row, we have a total of    choices, where    is the denominator of the fraction. Similarly, for the second entry in the first row, we have   choices, and so on. Therefore, the total number of choices for the first row is:
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