Set B = [1,1; 1, 1] and A = [zeros(2), eye(2); eye(2), B] and verify that B2 = O. (a)

Use MATLAB to generate random 4 4 matrices A and B. For each of the following, compute A1, A2, A3, and A4 as indicated and determine which of the matrices are equal (you can use MATLAB to test whether two matrices are equal by computing their difference): (a) A1 = A B, A2 = B A, A3 = (A_ B_ ) _, A4 = (B_ A_ ) _ (b) A1 = A_ B_, A2 = (A B) _, A3 = B_ A_, A4 = (B A) _ (c) A1 = inv(A B), A2 = inv(A) inv(B), A3 = inv(B A), A4 = inv(B) inv(A) (d) A1 = inv((A B) _ ), A2 = inv(A_ B_ ), A3 = inv(A_ ) inv(B_ ), A4 = (inv(A) inv(B)) _

Set n = 200 and generate an n n matrix and two vectors in Rn, both having integer entries, by setting A = floor(10 rand(n)); b = sum(A_ ) _; z = ones(n, 1) (Since the matrix and vectors are large, we use semicolons to suppress the printout.) (a) The exact solution of the system Ax = b should be the vector z. Why? Explain. One could compute the solution in MATLAB using the \ operation or by computing A1 and then multiplying A1 times b. Let us compare these two computational methods for both speed and accuracy. One can use MATLABs tic and toc commands to measure the elapsed time for each computation. To do this, use the commands tic, x = A\b; toc tic, y = inv(A) b; toc Which method is faster? To compare the accuracy of the two methods, we can measure how close the computed solutions x and y are to the exact solution z. Do this with the commands max(abs(x z)) max(abs(y z)) Which method produces the most accurate solution? (b) Repeat part (a), using n = 500 and n = 1000.

Set A = floor(10 rand(6)). By construction, the matrix A will have integer entries. Let us change the sixth column of A so as to make the matrix singular. Set B = A_ , A(:, 6) = sum(B(1 : 5, :)) _ (a) Set x = ones(6, 1) and use MATLAB to compute Ax. Why do we know that A must be singular? Explain. Check that A is singular by computing its reduced row echelon form. (b) Set B = x [1 : 6] The product AB should equal the zero matrix. Why? Explain. Verify that this is so by computing AB with the MATLAB operation . (c) Set C = floor(10 rand(6)) and D = B + C Although C _= D, the products AC and AD should be equal. Why? Explain. Compute AC and AD, and verify that they are indeed equal.

Construct a matrix as follows: Set B = eye(10) triu(ones(10), 1) Why do we know that B must be nonsingular? Set C = inv(B) and x = C(:, 10) Now change B slightly by setting B(10, 1) = 1/256. Use MATLAB to compute the product Bx. From the result of this computation, what can you conclude about the new matrix B? Is it still nonsingular? Explain. Use MATLAB to compute its reduced row echelon form.

Generate a matrix A by setting A = floor(10 rand(6)) and generate a vector b by setting b = floor(20 rand(6, 1)) 10 (a) Since A was generated randomly, we would expect it to be nonsingular. The system Ax = b should have a unique solution. Find the solution using the \ operation. Use MATLAB to compute the reduced row echelon form U of [A b]. How does the last column of U compare with the solution x? In exact arithmetic, they should be the same. Why? Explain. To compare the two, compute the difference U(:, 7) x or examine both using format long. (b) Let us now change A so as to make it singular. Set A(:, 3) = A(:, 1 : 2) [4 3]_ UseMATLAB to compute rref([A b]). How many solutions will the system Ax = b have? Explain. (c) Set y = floor(20 rand(6, 1)) 10 and c = A y Why do we know that the system Ax = c must be consistent? Explain. Compute the reduced row echelon form U of [ A c ]. How many solutions does the system Ax = c have? Explain. (d) The free variable determined by the echelon form should be x3. By examining the system corresponding to the matrix U, you should be able to determine the solution corresponding to x3 = 0. Enter this solution in MATLAB as a column vector w. To check that Aw = c, compute the residual vector c Aw. (e) Set U(:, 7) = zeros(6, 1). The matrix U should now correspond to the reduced row echelon form of ( A | 0 ). Use U to determine the solution of the homogeneous system when the free variable x3 = 1 (do this by hand), and enter your result as a vector z. Check your answer by computing A z. (f) Set v = w + 3 z. The vector v should be a solution of the system Ax = c. Why? Explain. Verify that v is a solution by using MATLAB to compute the residual vector c Av. What is the value of the free variable x3 for this solution? How could we determine all possible solutions of the system in terms of the vectors w and z? Explain.

Consider the graph V1 V8 V7 V5 V6 V2 V4 V3 (a) Determine the adjacency matrix A for the graph and enter it into MATLAB. (b) Compute A2 and determine the number of walks of length 2 from (i) V1 to V7, (ii) V4 to V8, (iii) V5 to V6, and (iv) V8 to V3. (c) Compute A4, A6, and A8 and answer the questions in part (b) for walks of lengths 4, 6, and 8. Make a conjecture as to when there will be no walks of even length from vertex Vi to vertex Vj . (d) Compute A3, A5, and A7 and answer the questions from part (b) for walks of lengths 3, 5, and 7. Does your conjecture from part (c) hold for walks of odd length? Explain. Make a conjecture as to whether there are any walks of length k from Vi to Vj based on whether i + j + k is odd or even. (e) If we add the edges {V3, V6}, {V5, V8} to the graph, the adjacency matrix B for the new graph can be generated by setting B = A and then setting B(3, 6) = 1, B(6, 3) = 1, B(5, 8) = 1, B(8, 5) = 1 Compute Bk for k = 2, 3, 4, 5. Is your conjecture from part (d) still valid for the new graph? (f) Add the edge {V6, V8} to the figure and construct the adjacency matrix C for the resulting graph. Compute powers of C to determine whether your conjecture from part (d) will still hold for this new graph.

In Application 1 of Section 4, the numbers of married and single women after 1 and 2 years were determined by computing the products AX and A2X for the given matrices A and X. Use format long and enter these matrices into MATLAB. Compute Ak and Ak X for k = 5, 10, 15, 20. What is happening to Ak as k gets large? What is the long-run distribution of married and single women in the town?

The following table describes a seven-stage model for the life cycle of the loggerhead sea turtle. (a) Enter L and x0 into MATLAB and use the command x50 = round(L50x0) to compute x50. Compute also the values of x100, x150, x200, x250, and x300. (b) Loggerhead sea turtles lay their eggs on land. Suppose that conservationists take special measures to protect these eggs and, as a result, the survival rate for eggs and hatchlings increases to 77 percent. To incorporate this change into our model, we need only change the (2,1) entry of L to 0.77. Make this modification to the matrix L and repeat part (a). Has the survival potential of the loggerhead sea turtle improved significantly? (c) Suppose that, instead of improving the survival rate for eggs and hatchlings, we could devise a means of protecting the small juveniles so that their survival rate increases to 88 percent. Use equations (1) and (2) from Application 2 of Section 4 to determine the proportion of small juveniles that survive and remain in the same stage and the proportion that survive and grow to the next stage. Modify your original matrix L accordingly and repeat part (a), using the new matrix. Has the survival potential of the loggerhead sea turtle improved significantly?

Set A = magic(8) and then compute its reduced row echelon form. The leading 1s should correspond to the first three variables x1, x2, and x3, and the remaining five variables are all free. (a) Set c = [1 : 8]_ and determine whether the system Ax = c is consistent by computing the re duced row echelon form of [A c]. Does the system turn out to be consistent? Explain. (b) Set b = [8 8 8 8 8 8 8 8]_ ; and consider the system Ax = b. This system should be consistent. Verify that it is by computing U = rref([A b]). We should be able to find a solution for any choice of the five free variables. Indeed, set x2 = floor(10 rand(5, 1)). If x2 represents the last five coordinates of a solution of the system, then we should be able to determine x1 = (x1, x2, x3)T in terms of x2. To do this, set U = rref([A b]). The nonzero rows of U correspond to a linear system with block form I V x1 x2 = c (1) To solve equation (1), set V = U(1 : 3, 4 : 8), c = U(1 : 3, 9) and use MATLAB to compute x1 in terms of x2, c, and V. Set x = [x1; x2] and verify that x is a solution of the system. 1

Set B = [1,1; 1, 1] and A = [zeros(2), eye(2); eye(2), B] and verify that B2 = O. (a) Use MATLAB to compute A2, A4, A6, and A8. Make a conjecture as to what the block form of A2k will be in terms of the submatrices I , O, and B. Use mathematical induction to prove that your conjecture is true for any positive integer k. (b) Use MATLAB to compute A3, A5, A7, and A9. Make a conjecture as to what the block form of A2k1 will be in terms of the submatrices I , O, and B. Prove your conjecture. 1

(a) The MATLAB commands A = floor(10 rand(6)), B = A_ A will result in a symmetric matrix with integer entries. Why? Explain. Compute B in this way and verify these claims. Next, partition B into four 33 submatrices. To determine the submatrices in MATLAB, set B11 = B(1 : 3, 1 : 3), B12 = B(1 : 3, 4 : 6) and define B21 and B22 in a similar manner, using rows 4 through 6 of B. (b) Set C = inv(B11). It should be the case that CT = C and B21T = B12. Why? Explain. Use the MATLAB operation _ to compute the transposes, and verify these claims. Next, set E = B21 C and F = B22 B21 C B21_ and use the MATLAB functions eye and zeros to construct L = I O E I , D = B11 O O F Compute H = L D L_ and compare H with B by computing H B. Prove that if all computations had been done in exact arithmetic, LDLT would equal B exactly.

Let A be a 4 3 matrix with a2 = a3. If b = a1 + a2 + a3, then the system Ax = b will have infinitely many solutions. 1

If E is an elementary matrix, then ET is also an elementary matrix. 1

The product of two elementary matrices is an elementary matrix. 1

If x and y are nonzero vectors in Rn and A = xyT , then the row echelon form of A will have exactly one nonzero row.