is an abelian group with respect to matrix multiplication.
1
Expert's answer
2021-06-08T08:48:01-0400
Let us check whether the set A={⎝⎛100a10bc1⎠⎞:a,b,c∈R} is an Abelian group with respect to matrix multiplication. It is sufficiently to prove that A is a subgroup of the general linear group GL3(R).
Let ⎝⎛100a110b1c11⎠⎞,⎝⎛100a210b2c21⎠⎞∈A.
Taking into account that
⎝⎛100a110b1c11⎠⎞⋅⎝⎛100a210b2c21⎠⎞=⎝⎛100a2+a110b2+a1c2+b1c2+c11⎠⎞∈A and
⎝⎛100a10bc1⎠⎞−1=⎝⎛100−a10ac−b−c1⎠⎞∈A, we conclude that A is a subgroup of GL3(R),
that is A is a group with respect to matrix multiplication.
Since ⎝⎛100110001⎠⎞⋅⎝⎛100010011⎠⎞=⎝⎛100110111⎠⎞ and
⎝⎛100010011⎠⎞⋅⎝⎛100110001⎠⎞=⎝⎛100110011⎠⎞, we conclude that the group A is not Abelian.
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