in Set Theory & Algebra edited by
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Let G be the set of all 2 x 2 symmetric, invertible matrices with real entries. Then with matrix multiplication, G is.

  1. An infinite group
  2. A finite group
  3. Not a group
  4. An abelian group
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All about Matrix Groups

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Infinite group as there are infinite elements in the real set and matrix multiplication satisfies the group properties - Closed, Associative, Identity, Inverse.

Commutativity is not satisfied and hence G is not Abelian.

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matrices are symmetric,matrix multiplication satisfies commutative property i think..
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The answer will be (c) not a group as it is not necessary that if A and b are symmetric then AB is also symmetric , so the closure property in this case will not be satisfies.

 

A counter example for this can be {{1,1},{1,1}}*{{1,2},{2,3}}={{3,5},{3,5}}

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In the example, u have taken a matrix of determinant zero. Please provide another example
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