University:
Carleton College
Course:
MATH 120 | Calculus 2
Academic year:

2024
Views:

77

Pages:

2
Author:

Nevaeh Bauer

Question 1 Write the system of linear equations in the form Ax = b and solve this matrix equation for x. x₁ + x₂ - 3x₃ = -1 -x₁ + 2x₂ = 1 x₁ - x₂ + x₃ = 2 Answer Firstly, switch to matrix form. [ 1 1 -3] [x₁] = [-1] [-1 2 0] [x₂] = [ 1] [ 1 -1 1] [x₃] = [ 2] Now, form the augmented matrix and by Gaussian elimination, reach the row-reduced echelon form. [ 1 1 -3 | -1 ] [-1 2 0 | 1 ] [ 1 -1 1 | 2 ] [ 1 1 -3 | -1 ] ~ [0 3 -3 | 0 ] R₂ + R₁ → R₂ [1 -1 1 | 2 ] [ 1 1 -3 | -1 ] ~ [0 3 -3 | 0 ] R₃ - R₁ → R₃ [0 -2 4 | 3] [ 1 1 -3 | -1 ] ~ [0 1 -1 | 0 ] (1/3)R₂ → R₂ [0 -2 4 | 3 ] [ 1 0 -2 | -1 ] ~ [ 0 1 -1 | 0 ] R₁ - R₂ → R₁ [ 0 0 2 | 3 ] R₃ + 2R₂ → R₃ [1 0 -2 | -1 ] ~ [0 1 -1 | 0 ] [0 0 1 | 1.5] (1/2)R₃ → R₃ [1 0 0 | 2 ] ~[0 1 0 | 1.5 ] R₁ + 2R₃ → R₁ [0 0 1 | 1.5] R₂ + R₃ → R₂ From here, we have the solution: x₁ = 2, x₂ = 1.5, x₃ = 1.5 Result: x₁ = 2, x₂ = 1.5, x₃ = 1.5 Question 2 What is a unit matrix? Answer 1 The unit matrix is every NxN square matrix made up of all zeros except for the elements of the main diagonal that are all ones. When the unit matrix is the product of two square matrices A and B, then A and B are said to be the inverses of each other. Answer 2 A matrix is called a unit or identity matrix if its diagonal elements are unit, that is 1, usually it is denoted by Iₙ. Question 3 Simplify by combining like terms 4x-6x Answer Subtract 6x from 4x. -2x - Answer Question 4 Find the characteristic polynomial of the matrices [ 1 2 1 ] A = [ 0 1 2 ] [-1 3 2] Answer Step 1 Given: [ 1 2 1 ] A = [ 0 1 2 ] [-1 3 2] To Find: Characteristic polynomial of the given matrices Step 2 Solution: [ 1 2 1] A = [ 0 1 2] [-1 3 2] [ 1-λ 2 1 ] A - λI = [ 0 1-λ 2 ] [-1 3 2-λ ] Characteristic polynomial of matrix A is given by |A – λI | = 0 [1 – λ 2 1 ] [0 1-λ 2 ] = 0 [-1 3 2-λ ] (1 – λ)[(1 – λ)(2 – λ) – 6] - 2[0 + 2] + 1[0 + (1 – λ)] = 0 (1 – λ)(2 - 3λ + λ² – 6) - 4 + 1 - λ = 0 (1 - λ)(-4 - 3λ + λ² ) - 3 - λ = 0 -4 - 3λ + λ² + 4λ + 3λ² - λ³ - 3 - λ= 0 After simplifying we get, λ³ - 4λ² + 7 = 0 Hence this is the required characteristic polynomial of the given matrix.