For a natural number n, let Cn= {(aij) e Mn(R): dij = Aki for all i, j, k, l such that j-inl – k}. (a) Exactly one of th

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For A Natural Number N Let Cn Aij E Mn R Dij Aki For All I J K L Such That J Inl K A Exactly One Of Th 1 (88.18 KiB) Viewed 24 times

For a natural number n, let Cn= {(aij) e Mn(R): dij = Aki for all i, j, k, l such that j-inl – k}. (a) Exactly one of the two matrices below is an element of C3. Which one? [1 mark] I A1 = 0 1 2 1 2 0 2 0 1 A2 = 0 1 2 2 0 1 1 2 0 (b) Let A be the matrix you chose in part (a). Compute A2. Check that A2 € C3, and write out a brief explanation (one or two sentences is enough) explaining how you checked. [2 marks] (c) Give a complete proof that C3 is a ring. You may assume that M3(R) is a ring. Hint: make as much use of that assumption as possible in your proof. [7 marks]