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18. Consider the matrices [2 1 X1 A = 2 2 -2 and x = x₂ -B 3 1 1 X3 a. Show that the equation Ax = x can be rewritten as (A-1)x= 0 and use this result to solve Ax = x for x. b. Solve Ax = 4x.

18. Consider the matrices [2 1 X1 A = 2 2 -2 and x = x₂ -B 3 1 1 X3 a. Show that the equation Ax = x can be rewritten as (A-1)x= 0 and use this result to solve Ax = x for x. b. Solve Ax = 4x.

College Algebra (MindTap Course List)
College Algebra (MindTap Course List)
12th Edition
ISBN:9781305652231
Author:R. David Gustafson, Jeff Hughes
Publisher:R. David Gustafson, Jeff Hughes
Chapter6: Linear Systems
Section6.6: Partial Fractions
Problem 62E: Match the rational expression on the left with the correct partial fraction decomposition form on...
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1.6 Questions 18 and 21 please on paper thank you
-2x₁ + x₂ = 0₂ 16. x₁2x₂ + 5x3 = b₁ 4x₁5x₂ + 8x3 = b₂ x₁2x₂x3 = b₁ -4x₁ + 5x₂ + 2x3 = b₂ -4x₁ + 7x₂ + 4x3 = b3 -3x₂ + 3x₂ 3x3 = bz x₁ - x₂ + 3x3 + 2x₁ = b₁ x₁ = b₂ -2x₁ + x₂ + 5x3 + -3x₁ + 2x₂ + 2x3 = x₁ = b₂ 4x₁3x₂ + x3 + 3x₁ = b₁ 18. Consider the matrices [2 1 2 X₁ A = 2 2 -2 and x = x₂ 3 1 1 X₂ a. Show that the equation Ax= x can be rewritten as (A-I)x= 0 and use this result to solve Ax = x for x. b. Solve Ax = 4x. In Exercises 19-20, solve the matrix equation for X. [1 -1 1 [2 -1 5 7 8 19. 2 0 -3 0 1 0 5 -7 2 1 -2 15. 17. 32 0X = 4 243 -1 0 1 4 3 2 1 20. 0 -1 -1 X = 6 7 8 9 1 1 -4 1 3 79 Working with Proofs 21. Let Ax = 0 be a homogeneous system of n linear equations in n unknowns that has only the trivial solution. Prove that if k is any positive integer, then the system Akx = 0 also has only the trivial solution. 22. Let Ax = 0 be a homogeneous system of n linear equations in n unknowns, and let Q be an invertible nxn matrix. Prove that Ax = 0 has only the trivial solution if and only if (QA)x= 0 has only the trivial solution. 23. Let Ax = b be any consistent system of linear equations, and let x₁ be a fixed solution. Prove that every solution to the sys- c. If A and B are nx BA=In. d. If A and B are rov systems Ax = 0 ar e. Let A be an nx matrix. If x is a s then Sx is a soluti f. Let A be an nx n a unique solution matrix. g. Let A and B ber invertible, then Working with Techn T1. Colors in print me vision screens are i models." For examp mixing percentages the YIQ model (us by mixing percenta of a chrominance f conversion from t plished by the ma Y 1 QI What matrix wor RGB model? [1 -2 A = 4 5 0 3 Solve the linea: the method of l T2. Let
Transcribed Image Text:-2x₁ + x₂ = 0₂ 16. x₁2x₂ + 5x3 = b₁ 4x₁5x₂ + 8x3 = b₂ x₁2x₂x3 = b₁ -4x₁ + 5x₂ + 2x3 = b₂ -4x₁ + 7x₂ + 4x3 = b3 -3x₂ + 3x₂ 3x3 = bz x₁ - x₂ + 3x3 + 2x₁ = b₁ x₁ = b₂ -2x₁ + x₂ + 5x3 + -3x₁ + 2x₂ + 2x3 = x₁ = b₂ 4x₁3x₂ + x3 + 3x₁ = b₁ 18. Consider the matrices [2 1 2 X₁ A = 2 2 -2 and x = x₂ 3 1 1 X₂ a. Show that the equation Ax= x can be rewritten as (A-I)x= 0 and use this result to solve Ax = x for x. b. Solve Ax = 4x. In Exercises 19-20, solve the matrix equation for X. [1 -1 1 [2 -1 5 7 8 19. 2 0 -3 0 1 0 5 -7 2 1 -2 15. 17. 32 0X = 4 243 -1 0 1 4 3 2 1 20. 0 -1 -1 X = 6 7 8 9 1 1 -4 1 3 79 Working with Proofs 21. Let Ax = 0 be a homogeneous system of n linear equations in n unknowns that has only the trivial solution. Prove that if k is any positive integer, then the system Akx = 0 also has only the trivial solution. 22. Let Ax = 0 be a homogeneous system of n linear equations in n unknowns, and let Q be an invertible nxn matrix. Prove that Ax = 0 has only the trivial solution if and only if (QA)x= 0 has only the trivial solution. 23. Let Ax = b be any consistent system of linear equations, and let x₁ be a fixed solution. Prove that every solution to the sys- c. If A and B are nx BA=In. d. If A and B are rov systems Ax = 0 ar e. Let A be an nx matrix. If x is a s then Sx is a soluti f. Let A be an nx n a unique solution matrix. g. Let A and B ber invertible, then Working with Techn T1. Colors in print me vision screens are i models." For examp mixing percentages the YIQ model (us by mixing percenta of a chrominance f conversion from t plished by the ma Y 1 QI What matrix wor RGB model? [1 -2 A = 4 5 0 3 Solve the linea: the method of l T2. Let
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