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Problem 5 (Algebraic Properties Of Square Matrices)
Let $\text{M}_2(\mathbb{Q})$ bet the set of 2 by 2 matrices with entries in $\mathbb{Q}$, together with the usual properties of matrix addition and matrix multiplication. Recall that $\text{GL}(2,\mathbb{Q})$ is the set of invertible 2 by 2 matrices with entries in $\mathbb{Q}$.
- Which of the properties of being a field does $\text{M}_2(\mathbb{Q})$ not satisfy? Give an example of each property that is not satisfied.
- Which of the properties of being a field does $\text{GL}(2,\mathbb{Q})$ not satisfy? Give an example of each property that is not satisfied.
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