Write the elements of matrix r in terms of real-numbers r1

Question

Question #1. Consider a sequence of non-zero real-numbers r1, r2,............rN such that no two numbers are equal. Using these numbers, we create an (N x N) square matrix R such that the (i, -j)-th element of R is given by

a_ij = a(i,j) = (i,j)-th element of R = rk,

where k = min(i,j), i = 1, 2, ..., N;j = 1, 2, ..., N. For example, min(2, 5) = 2. Similarly, min(78, 49) = 49 => a(78, 49) = r49. In general

aij = a(i,j)= (0)-th element of R = r₂ k = min(i, j).

In other words, given i and j, find k first, and then find a(i,j) = rt.

A. Write the elements of matrix R in terms of real-numbers r1, r2,..............rN. Clearly, show at least the top 4 x 4 part and all the elements on the four corners.

B. Is this a symmetric matrix?

C. Carry out appropriate EROs to reduce the matrix R to its echlon form.

D. Carry out appropriate EROs to reduce the matrix R to its row echlon form.

E. Carry out appropriate EROs to reduce the matrix R to its reduced row echlon form.

F. It is claimed that a unique solution always exists for the linear system Ri = b. Do you agree? Give justification.

G. Solve the system when N = 10. rk = k. k = 1, 2, ..., 10, and b = [1 1 1 ... I ]^T.