FINAL Here is the nal. Try to adhere to conventions as much as possible. This will allow us to grade more accurately and eciently. Problem 5 covers material from this Wednesday up until next Wednesday. Please email me if there are any errors. Good luck! 1: 2: 3: 4: 5: This is the point distribution. 5 4.5 4 4.5 7 Problem 1. Let M = a c b be an invertible matrix. d a) If a = 0, nd elementary matrices E1 , . . . , En such that En E1 M = I (Choose your matrices as we have been doing in class-that is, in such a way that the matrices E1 , . . . , En are in the order Type L/T, Type D, Type U. The product, En E1 is therefore in the reverse of this order). If your answer involves fractions, tell me why the denominator is nonzero in each fraction. b) If a = 0, nd elementary matrices F1 , . . . , Fm such that Fm F1 M = I (Choose your matrices as we have been doing in class-that is, in such a way that the matrices F1 , . . . , Fm are in the order Type L/T, Type D, Type U). If your answer involves fractions, tell me why the denominator is nonzero in each fraction. d b = Fm F1 by multiplying out the c a matrices you found in parts (b) and (c). c) Show that En E1 = 1 adbc Problem 2. For an n x n matrix, M , let F(M ) be the determinant formula involving permutations discussed in class (it is called \"big\" formula in book.) Use the denition of R1 . F to prove that if M is written in terms of its rows: M = . then: . Rn 1 2 FINAL R1 R1 . . . . . . a) F Ri + Ri = F Ri + F Ri for any length n vector, Ri . . . . . . . . . . Rn Rn Rn R1 R1 . . . . . . c(Ri ) = c F Ri for any constant, c. b) F . . . . . . Rn Rn R1 R1 . . . . . . Ri Rj . . c) F . = F . (this is dicult). . . Rj Ri . . . . . . Rn Rn R1 . . . Hint for (c): First show that the permutation corresponding to switching i and j, that is the permutation: ij = (1, . . . , i 1, j, i + 1, . . . , j 1, i, j + 1, . . . , n), has an odd inversion number. For any permutation, we dene sgn() = (1)inv() . (This means sgn(ij ) = 1.) Show that for any permutations and , we have sgn( ) = sgn()sgn(). Therefore you may conclude that sgn( ij ) = sgn()sgn(ij ) = sgn(). Now, use that sgn( ij ) = sgn(), and that 1 ij = ij to show (c). For half credit on (a), (b), or (c) you can use n = 3, i = 2, and j = 3. Problem 3. Let A be an n x n matrix. Suppose A is invertible and symmetric. Show that A1 is also symmetric. Problem 4 (a) Let M be an n x n matrix. Suppose M is upper diagonal. Suppose that its diagonal entries are d1 , . . . , dn . What is Det(M ) in terms of the di 's? Prove your answer directly (without using the formula F; equivalently without row/column expansion). Hints: 1. If M is not invertible what do you know about the diagonal entries (previous HW) and about Det(M )? 2. If M is invertible, we can nd elementary matrices such that En E1 M = I. 3. What (type L/T matrices) are used in step 1? Does M change during step 1? FINAL 3 4. What (type D matrices) are used in step 2? 5. What is Det(En E1 )? 6. How does Det(M ) relate to Det(En E1 )? (b) Let N be an n x n matrix. Suppose N is lower diagonal. Suppose that its diagonal entries are f1 , . . . , fn . What is Det(N ) in terms of the fi 's? Prove your answer. Hint: Use part (a) and what you know about determinants of transposes. Problem 5 2015 2 0 3 . Write A as a product of matrices: A = P DP 1 (a) Let A = 0 1 2 5 2 4 for some intertible matrix, P , and some diagonal matrix, D. Compute the entries of P and D explicitly, but don't compute the entries of P 1 . Show all your work. Then, use a computer to evaluate the product P DP 1 . n 1 2 3 4 5 2 1 1 1 1 4 2 3 = I. (b) Consider the equation: 3 1 4 1 2 5 1 5 1 3 1 1 Does n = 2 make this equation true? Is there any positive integer n that makes this equation true? How do you know this without using a computer or computing the determinant? c) Are there real numbers a and b such that 5a2 + 10b2 = 14ab