Please just 26 and 27 1.6 EXERCISESThe matrices and vectors listed in Eq. (3) are used inseveral of the exercises that follow.31121-⠀ -⠀A = 4 7B = 7 4 3266012 40--[!]C =613524 203 6*=[28]E=3D=[²4].*=[¦}]--[0) 46)+-[3](3)Exercises 1-25 refer to the matrices and vectors inEq. (3). In Exercises 1-6, perform the multiplicationsto verify the given equality or nonequality.1. (DE)F = D(EF)3. DE # ED2. (FE)D = F(ED)4. EF # FE6.3Fu = 7Fv5. Fu = FvIn Exercises 7-12, find the matrices.7. AT8. DT10. ATC11. (FV)20. ||DV||23. || Ful9. ETF12. (EF)vIn Exercises 13-25, calculate the scalars.13. u7v14. v¹ Fu16. v¹ Fv17. u u19. u22. ||u - v25. ||(D-E)u||26. Let A and B be (2x2) matrices. Prove or find a coun-terexample for this statement: (AB)(A + B) =A² - B².15. v Dv18. v'v21. ||Au||24. ||FV||27. Let A and B be (2 x 2) matrices such that A² = ABand A # O. Can we assert that, by cancellation,A = B? Explain.28. Let A and B be as in Exercise 27. Find the flaw inthe following proof that A = B.Since A² = AB, A² - AB = O. Factoringyields A(A - B) = O. Since A # O, it follows thatA - 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 andB are symmetric, but AB is not symmetric. [Hint:(AB)¹ = B¹A¹ = BA.]31. Let A and B be (n x n) symmetric matrices. Givea necessary and sufficient condition for AB to besymmetric. [Hint: Recall Exercise 30.]32. Let G be the (2 x 2) matrix that follows, and con-sider any vector x in R² where both entries are notsimultaneously zero:--[G]-[R]G = [ ¦ ¦ ] ; x = [|x₁| + x₂ 0.Show that x¹Gx 0. [Hint: Write x¹Gx as a sumof squares.]33. Repeat Exercise 32 using the matrix D in Eq. (3) inplace of G.X234. For F in Eq. (3), show that xFx ≥ 0 for all x inR². Classify those vectors x such that xFx=0.If x and y are vectors in R", then the product x'y is of-ten called an inner product. Similarly, the product xyis often called an outer product. Exercises 35-40 con-cern outer products; the matrices and vectors are given inEq. (3). In Exercises 35-40, form the outer products.35. uv38. u(Ev)41. Let a and b be given by36. u(Fu)39. (Au) (Av)37. v(Ev)40. (Av) (Au)-----and b=-[³]a) Find x in R² that satisfies both x a = 6 andx¹b = 2.=b) Find x in R² that satisfies both x²(a + b) = 12and x¹a = 2.42. Let A be a (2 x 2) matrix, and let B and C be givenby13*-[B]-[B]and C=a) If A¹ + B = C, what is A?234 5

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Please just 26 and 27 1.6 EXERCISESThe matrices and vectors listed in Eq. (3) are used inseveral of the exercises that follow.31121-⠀ -⠀A = 4 7B = 7 4 3266012 40--[!]C =613524 203 6*=[28]E=3D=[²4].*=[¦}]--[0) 46)+-[3](3)Exercises 1-25 refer to the matrices and vectors inEq. (3). In Exercises 1-6, perform the multiplicationsto verify the given equality or nonequality.1. (DE)F = D(EF)3. DE # ED2. (FE)D = F(ED)4. EF # FE6.3Fu = 7Fv5. Fu = FvIn Exercises 7-12, find the matrices.7. AT8. DT10. ATC11. (FV)20. ||DV||23. || Ful9. ETF12. (EF)vIn Exercises 13-25, calculate the scalars.13. u7v14. v¹ Fu16. v¹ Fv17. u u19. u22. ||u - v25. ||(D-E)u||26. Let A and B be (2x2) matrices. Prove or find a coun-terexample for this statement: (AB)(A + B) =A² - B².15. v Dv18. v'v21. ||Au||24. ||FV||27. Let A and B be (2 x 2) matrices such that A² = ABand A # O. Can we assert that, by cancellation,A = B? Explain.28. Let A and B be as in Exercise 27. Find the flaw inthe following proof that A = B.Since A² = AB, A² - AB = O. Factoringyields A(A - B) = O. Since A # O, it follows thatA - 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 andB are symmetric, but AB is not symmetric. [Hint:(AB)¹ = B¹A¹ = BA.]31. Let A and B be (n x n) symmetric matrices. Givea necessary and sufficient condition for AB to besymmetric. [Hint: Recall Exercise 30.]32. Let G be the (2 x 2) matrix that follows, and con-sider any vector x in R² where both entries are notsimultaneously zero:--[G]-[R]G = [ ¦ ¦ ] ; x = [|x₁| + x₂ > 0.Show that x¹Gx > 0. [Hint: Write x¹Gx as a sumof squares.]33. Repeat Exercise 32 using the matrix D in Eq. (3) inplace of G.X234. For F in Eq. (3), show that xFx ≥ 0 for all x inR². Classify those vectors x such that xFx=0.If x and y are vectors in R&#34;, then the product x'y is of-ten called an inner product. Similarly, the product xyis often called an outer product. Exercises 35-40 con-cern outer products; the matrices and vectors are given inEq. (3). In Exercises 35-40, form the outer products.35. uv38. u(Ev)41. Let a and b be given by36. u(Fu)39. (Au) (Av)37. v(Ev)40. (Av) (Au)-----and b=-[³]a) Find x in R² that satisfies both x a = 6 andx¹b = 2.=b) Find x in R² that satisfies both x²(a + b) = 12and x¹a = 2.42. Let A be a (2 x 2) matrix, and let B and C be givenby13*-[B]-[B]and C=a) If A¹ + B = C, what is A?234 5

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26. Let A and B be (2x2) matrices. Prove or find a coun- terexample for this statement: (A-B)(A + B) = A²-B².