Math 4330 Sec. 1, Matlab Assignment # 1, March 24, 2006 Name 1 ...

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1. Enter the matrices and carry out the following: A = ASSIGNMENT 1 [ ] 2 4 , B = 1 3 (a) Verify that (A + B) + C = A + (B + C) [ ] −2 1 , C = 0 4 [ ] 3 1 2 1 (A + B) + C = A + (B + C) = (b) Verify that (AB)C = A(BC) (AB)C = A(BC) = (c) Decide whether AB is equal to BA AB = BA = (d) Find (A + B) 2 , (A 2 + 2AB + B 2 ) and (A 2 + AB + BA + B 2 ) (A + B) 2 = (A 2 + 2AB + B 2 ) = (e) Find A 2 − B 2 , (A − B)(A + B) A 2 − B 2 = (A − B)(A + B) = 2. Enter and do the following: A = [ ] 1/2 1/2 , B = 1/2 1/2 [ 1/2 ] −1/2 −1/2 1/2 (a) Compute A 2 = , A 3 = , etc. Can you say what A n will be Explain why this is true. (b) Compute B 2 = , Can you explain why this is true. What does this tell you about matrix multiplication that is different from squaring numbers
(c) Find AB = and BA = . What do you learn from this that is not true for multiplication of numbers (hint: if a and b are real numbers and ab = 0, then a = 0 or b = 0). 3. Find the inverse of the matrices (if they exist) and check that the result is correct by multiplying the matrix times its inverse. [ ] −1 1 (a) A = , A 1 0 −1 = , (b) B = [ ] 2 5 , B 1 3 −1 = , ⎡ ⎤ 2 0 5 (c) C = ⎣0 3 0⎦, C −1 = , 1 0 3 ⎡ ⎤ −1 −3 −3 (d) D = ⎣ 2 6 1 ⎦, D −1 = , 3 8 3 [ ] 1 −1 4. Consider the A δ = depending on a parameter δ. The determinant of A 1 −1 + δ δ is δ so when δ is small it is clear that the close to having a zero eigenvalue. This does not necessarily mean the matrix has a large condition number but it raises a red flag. Using Maple I compute the condition number using the command ConditionNumber(A_d); to get (∣ ) ∣∣∣ −1 + δ max δ ∣ + (|δ|)−1 , 2 (|δ|) −1 max (2, 1 + |−1 + δ|) Notice here is an example where Maple is preferably since I want the answer in terms of δ. [ 1 Set x = and set b 1] δ = A d x. We consider δ = 10 −1 and δ = 10 −5 . (a) For each δ find the eigenvalues and condition numbers of A δ . (b) For each δ solve A δ x δ = b. (c) Consider the error obtained between x and x δ . Now let us perturb the right hand side a little bit. , , ,
Page 1 and 2: Math 4330 Sec. 1, Matlab Assignment
Page 3 and 4: (c) A has an inverse inv(A) inv(B)
Page 5: On the left we solve the resulting
Page 9: (b) Compute det(A) (c) Compute rref

Math 4330 Sec. 1, Matlab Assignment # 1, March 24, 2006 Name 1 ...

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