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Let A be a square matrix all of whose...

Let A be a square matrix all of whose entries are integers. Then which one of the following is true? (1) If `d e t A""=+-1,""t h e n""A^(1)` exists but all its entries are not necessarily integers (2) If `d e t A!=""+-1,""t h e n""A^(1)` exists and all its entries are non-integers (3) If `d e t A""=+-1,""t h e n""A^(1)` exists and all its entries are integers (4) If `d e t A""=+-1,""t h e n""A^(1)` need not exist

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To solve the given problem, we need to analyze the properties of a square matrix \( A \) with integer entries and its determinant. We will evaluate each option provided in the question. ### Step-by-Step Solution: 1. **Understanding the Determinant**: - A square matrix \( A \) has a determinant denoted as \( \text{det}(A) \). If \( \text{det}(A) \neq 0 \), then the inverse of \( A \), denoted as \( A^{-1} \), exists. 2. **Evaluating the Options**: ...
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