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Let ๐บ = [๐‘Ž 00 0] : ๐‘Ž ๐‘–๐‘  ๐‘Ž ๐‘›๐‘œ๐‘› โˆ’ ๐‘ง๐‘’๐‘Ÿ๐‘œ ๐‘Ÿ๐‘’๐‘Ž๐‘™ ๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ } . hho that ๐บ is a oommutativegroup under multiplioation

Question

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Solution

The question seems to be asking to prove that the set G, consisting of matrices of the form [a 00 0] where a is a non-zero real number, forms a commutative group under multiplication. However, the question is not clearly written and contains several typographical errors. Here's a step-by-step guide to prove it:

  1. Identity: The identity element in the group under multiplication is the matrix [1 00 0]. Any matrix in G when multiplied with this identity matrix gives the same matrix, hence the identity exists.

  2. Closure: If we take any two matrices [a 00 0] and [b 00 0] in G and multiply them, we get another matrix [ab 00 0] which is also in G. Hence, the set G is closed under multiplication.

  3. Inverse: For every matrix [a 00 0] in G, there exists an inverse [1/a 00 0] such that their multiplication gives the identity matrix [1 00 0]. Hence, every element has an inverse in G.

  4. Commutativity: For any two matrices [a 00 0] and [b 00 0] in G, the order of multiplication does not matter. [a 00 0] * [b 00 0] = [ab 00 0] = [b 00 0] * [a 00 0]. Hence, the group is commutative.

Therefore, G is a commutative group under multiplication.

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