Notes:

1. This CA is worth 10% of the grade. It will be graded out of 100.
2. Your submission will be treated as a declaration by you that you have not consulted a human person (another student registered for this module or anyone else) for your submission.

Date of distribution: September 08, 2016.
Due date: September 28, 2016 at the end of the lecture; No extensions or electronic submissions are allowed.

Question #1. Consider a sequence of non-zero real-numbers r₁, r₂, ..., r₁₀₀ such that no two numbers are equal except r₁₀₀ = r₉₉. Using these numbers, we create an (100 x 100) square matrix R such that the (i, j)-th element of R is given by

a_i,j = a(i, j) = (i, j)-th element of R = r_k,

where k = min(i, j), i = 1, 2, ..., N; j = 1, 2, ..., N. For example, min(2, 5) = 2.

Similarly, min(78, 49) = 49 ⇒ a78, 49 = a(78, 49) = r49.

In general

ai, j = a(i, j) = (i, j)-th element of R = rk, k = min(i, j).

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

A. Write the elements of matrix R in terms of real-numbers r1, r2, ..., r100. 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 RT to its echlon form.

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

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

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

G. Solve the system when r_k = k, k = 1, 2, ..., 99, and b = [1 1 1 ... 1]^T.

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 either from top down or bottom up.