The set of all 2 times 2 matrices over the re | vedclass

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(D) group $(G, \cdot)$ must satisfy four axioms: closure,associativity,identity,and inverse. For the set of all $2 	imes 2$ matrices under multiplica

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Inverse axiom may not be satisfied

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(D) group $(G, \cdot)$ must satisfy four axioms: closure,associativity,identity,and inverse. For the set of all $2 	imes 2$ matrices under multiplication,the identity element is the identity matrix $I = \begin{bmatrix} 1 & 0 \ 0 & 1 \end{bmatrix}$. However,for an element to have an inverse,its determinant must be non-zero. Since there exist $2 	imes 2$ matrices with a determinant of $0$ (singular matrices),these matrices do not have a multiplicative inverse. Therefore,the inverse axiom is not satisfied for all elements in the set,making it not a group.

The set of all $2 \times 2$ matrices over the real numbers is not a group under matrix multiplication because

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