Question 26
4.6 Algebraic Properties of Matrix Operations 367 4.6 EXERCISES The matrices and vectors listed in Eq. (3) are used in Since A? = AB, A2 - AB = O. Factoring several of the exercises that follow. yields A(A - B) = O. Since A * O. it follows that A - B = O. Therefore, A = B. 29. Two of the six matrices listed in Eq. (3) are symmet- ric. Identify these matrices. 30. Find (2 x 2) matrices A and B such that A and B are symmetric, but AB is not symmetric. [Hint: (AB) - B A - BA.] 31. Let A and B be (n x n) symmetric matrices. Give a necessary and sufficient condition for AB to be symmetric. [Hint: Recall Exercise 30.] 32. Let G be the (2 x 2) matrix that follows, and con- E-[ 2 8]. F-[: ] sider any vector x in R' where both entries are not simultaneously zero: Lxil + 1x21 2 0 . Show that x Gx > 0. [Hint: Write x Gx as a sum (3) of squares.] Exercises 1-25 refer to the matrices and vectors in 33. Repeat Exercise 32 using the matrix D in Eq. (3) in Eq. (3). In Exercises 1-6, perform the multiplications place of G. to verify the given equality or nonequality. 34. For F in Eq. (3), show that x Fx 2 0 for all x in 1. (DE) F = D(EF) 2. (FE) D = F(ED) R2. Classify those vectors x such that x Fx = 0. 3. DE # ED 4. EF + FE If x and y are vectors in R", then the product x y is of- ten called an inner product. Similarly, the product xy 5. Fu = Fv 6. 3Fu = 7Fv is often called an outer product. Exercises 35-40 con- In Exercises 7-12, find the matrices. cern outer products; the matrices and vectors are given in Eq. (3). In Exercises 35-40, form the outer products. 7. AT 8. DT 9. ETF 35. uvT 36. u(Fu) 37. (Ev) 10. ATC 11. (Fv)? 12. (EF)v 38. u(Ev) T 39. (Au)(AV)T 40. (Av) (Au) In Exercises 13-25, calculate the scalars. 41. Let a and b be given by 13. uv 14. v Fu 15. v DV 16. v Fv 17. u u 18. vv [:] and b = [ ]. 19. Ju 20. DVI 21. | Aull 22. lu - vil 23. | Full 24. IFVI a) Find x in R2 that satisfies both x a = 6 and 25. (D - E)ull x b = 2. 26. Let A and B be (2x2) matrices. Prove or find a coun- b) Find x in R2 that satisfies both x(a + b) = 12 terexample for this statement: (A - B)(A + B) = and xa = 2. A2 - B2. 42. Let A be a (2 x 2) matrix, and let B and C be given 27. Let A and B be (2 x 2) matrices such that A2 = AB by and A # O. Can we assert that, by cancellation, A = B? Explain. and c = [ 4 3 ]. 28. Let A and B be as in Exercise 27. Find the flaw in the following proof that A = B. a) If AT + B = C, what is A? Hw2(P102).jpg Sep
