Consider a sequence of positive real-numbers r ₁ , r ₂ ,…..,r _N , such that no two numbers are the same. Using these numbers, we create two (N x N) square matrices R and S such that R is a lower triangular matrix and S is an upper triangular matrix. The (i, j)-th element of R and S are given by

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A. Write the elements of matrices R and S in terms of real-numbers ri, r2, ..., rN. Clearly, show at least the top 4 x 4 part and all the elements on the four corners. (10) B. Let U = R+ S. Write the top 4 x 4 part of U. Is U a symmetric matrix? (10) C. Carry out appropriate EROS to reduce the matrix R to its echlon form. (10) D. Carry out appropriate EROS to reduce the matrix R to its row echlon form. (10) E. Carry out appropriate EROS to reduce the matrix R to its reduced row echlon form. (10) F. Carry out appropriate EROS to reduce the matrix S to its echlon form. (10) G. Carry out appropriate EROS to reduce the matrix S to its row echlon form. (10) H. Carry out appropriate EROS to reduce the matrix S to its reduced row echlon form. (10) I. It is claimed that a unique solution always exists for the linear system S x b. Do you agree? Give justification. (10) J. Solve the system Sx b when N= 5, r k, k = 1, 2,.., 5, and b [1 11 ... 1]'. (10) HINT: We introduced four technical words: 'forward', 'elimination', 'backward', 'substitution'. See if a mix of these will be useful here. Try EROS such that you do forward elimination from the bottom. A. Write the elements of matrices R and S in terms of real-numbers ri, r2, ..., rN. Clearly, show at least the top 4 x 4 part and all the elements on the four corners. (10) B. Let U = R+ S. Write the top 4 x 4 part of U. Is U a symmetric matrix? (10) C. Carry out appropriate EROS to reduce the matrix R to its echlon form. (10) D. Carry out appropriate EROS to reduce the matrix R to its row echlon form. (10) E. Carry out appropriate EROS to reduce the matrix R to its reduced row echlon form. (10) F. Carry out appropriate EROS to reduce the matrix S to its echlon form. (10) G. Carry out appropriate EROS to reduce the matrix S to its row echlon form. (10) H. Carry out appropriate EROS to reduce the matrix S to its reduced row echlon form. (10) I. It is claimed that a unique solution always exists for the linear system S x b. Do you agree? Give justification. (10) J. Solve the system Sx b when N= 5, r k, k = 1, 2,.., 5, and b [1 11 ... 1]'. (10) HINT: We introduced four technical words: 'forward', 'elimination', 'backward', 'substitution'. See if a mix of these will be useful here. Try EROS such that you do forward elimination from the bottom.