Fsc Part 1 Mathematics (Complete Solution)

Q10
Prove that all 2x2 non-singular matrices over the real field form a non-abelian group under multiplication.

SOLUTION:

Let G be the all non-singular 2x2 matrices over the real field.

(i) Let A,B∈G then A2x2xB2x2=C2x2∈ G

Thus closure law holds in G under multiplication

(ii) Associative law in matrices of same order under multiplication holds therefore for A,B,C∈G

Ax(BxC)=(AxB)xC

(iii) Since identity matrix of order two is also a no singular matrix.

(iv) As we know for any two matrices A,B∈G, AB≠BA in general.

Therefore commutative law does not holds in G under multiplication.

Hence the set of all 2x2 non-singular matrices over a real field is a non-abelian group under multiplication

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