 # 3. [3pt] Given is a tridiagonal matrix, i.e., a matrix with nonzero...

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3. [3pt] Given is a tridiagonal matrix, i.e., a matrix with nonzero entries only in the diagonal, and the first upper and lower subdiagonals: a1 C1 b1 a2 C2 A bn-2 an-1 Cn-1 bn-1 an Assuming that A has an LU decomposition A = LU with 1 e1 f1 1 1 L , en-1 fn-1 dn-1 en derive recursive expressions for di, ei and fi. 1 4. [4pt] For a given dimension n, fix some k: with 1 V k < n. Now let L € Rnxn be a non-singular lower triangular matrix and let the vector b € Rn be such that bi - 0 for i = 1. , 2 , ,k. , (a) Let the vector y € Rn be the solution of Ly = b. Show, by partitioning L into blocks, that yj = 0 for j = 1, , 2. k. , (b) Use this to give an alternative proof of Theorem 2.1(iv), i.e., that the inverse of a non-singular lower triangular matrix is itself lower triangular.

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