In this exercise you will see how matrix multiplication can be used to get a nice...

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In this exercise you will see how matrix multiplication can be used to get a nice proof of the major identities that play a role in the extended Euclidean algorithm. We consider two integers a, b e Z and the sequences ri, qi, Xi, y; that come from applying the extended Euclidean algorithm to a, b. Suppose that rn+1 = 0 (i.e., rn = gcd(a, b)). - (2). (8) for k= 2,..., n + 1. (b) Define xo = 1, x₁ = 0 and yo = 0, y₁ = 1 and x₁ = xi-2 - qixi-1 and yi = yi-2 — qiyi-1. Prove that (a) For i = 2,..., n + 1, define A; = (c) Prove that . Prove that |Xi-1 Ni-1 -1|Xi-2 Xi Yi and conclude that xia+yib = ri. Ni-2\ = A₂ Xi-1 Yi-1) (**)(0) = (2) Xi Yi = A₂... Ak Ik-1 rk for i= 2,..., n + 1. for i= 1,..., n+1 In this exercise you will see how matrix multiplication can be used to get a nice proof of the major identities that play a role in the extended Euclidean algorithm. We consider two integers a, b e Z and the sequences ri, qi, Xi, y; that come from applying the extended Euclidean algorithm to a, b. Suppose that rn+1 = 0 (i.e., rn = gcd(a, b)). - (2). (8) for k= 2,..., n + 1. (b) Define xo = 1, x₁ = 0 and yo = 0, y₁ = 1 and x₁ = xi-2 - qixi-1 and yi = yi-2 — qiyi-1. Prove that (a) For i = 2,..., n + 1, define A; = (c) Prove that . Prove that |Xi-1 Ni-1 -1|Xi-2 Xi Yi and conclude that xia+yib = ri. Ni-2\ = A₂ Xi-1 Yi-1) (**)(0) = (2) Xi Yi = A₂... Ak Ik-1 rk for i= 2,..., n + 1. for i= 1,..., n+1