Consider a sequence of 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
ai,j = rk, where k = min(i, j), for i ≥ j.
The elements ai,j are arbitrary for i < j. The indices i and j take values over the range 1, 2, ..., N.
A. Let N = 5. Write elements of matrix (5 ! 5) R in terms of real-numbers r1, r2, ..., rN and other arbitrary values. Clearly, show the top 4 ! 4 part and all the elements on four corners. [20 points]
B. Is R a symmetric matrix? Give reason for your answer. If the answer is no, what conditions should the elements above the main diagonal satisfy to make R a symmetric matrix. [10 points]
C. Carry out appropriate EROs to reduce the matrix R to its echlon form. Is echlon form unique? Give at least two different versions of echlon form if your answer is no. [30 points]
D. Starting from the echlon form in C, carry out EROs to reduce the matrix R to its row echlon form. Is row echlon form unique? Give at least two different versions of row echlon form if your answer is no. [10 points]
E. Starting from the row echlon form in D, describe EROs to reduce the matrix R to its reduced row echlon form. There is no need to carry out these EROs. [10 points]
F. State conditions for a unique solution to exist for the linear system Rx = b in terms of its echlon form derived in C. Give justification. [10 points]
G. Now, b = [0 0 ... 0]T, an all-0 column vector. Study the linear system Rx = b. In terms of echlon form derived in C, state conditions for multiple solutions to exist. [10 points]
H. Consider ERO1, namely exchange of two rows of matrix A. Let B be the matrix obtained after the ERO1 is performed on A. Find a matrix T such that B = T A. [OPTIONAL]