Answer to Question #108187 in Abstract Algebra for Garima Ahlawat

Question #108187
Is R = {[a a+b
a+b b ] |a,b belongs to Z} a subring of M2(Z)? Why or why not?
1
Expert's answer
2020-04-08T11:45:35-0400

"\\begin{pmatrix}\n 1 & 3 \\\\\n 3 & 2\n\\end{pmatrix}\\in R" and "\\begin{pmatrix}\n 1 & 2 \\\\\n 2 & 1\n\\end{pmatrix}\\in R", but "\\begin{pmatrix}\n 1 & 3 \\\\\n 3 & 2\n\\end{pmatrix}\\begin{pmatrix}\n 1 & 2 \\\\\n 2 & 1\n\\end{pmatrix}=\\begin{pmatrix}\n 7 & 5 \\\\\n 7 & 8\n\\end{pmatrix}\\not\\in R". So "R" is not a subring of "M_2(\\mathbb Z)".


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