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Page < 2 of 2 - ZOOM + 3) a) Find the characteristic equation of the matrix b) Find the eigenvalues of A. A = [ 12 13³] c) Find each eigenvector corresponding to each eigenvalue of A. 4) a) Find a nonsingular matrix P such that P-1 AP is a diagonal matrix for the following matrix: A = 1 2 -21 -2 5 -2 -6 6 -3] b) Verify that P-1AP is a diagonal matrix with the eigenvalues on the main diagonal. Page < 1 of 2 - ZOOM + 1) Answer the following questions by circling TRUE or FALSE (No explanation or work required). −1 0 01 i) If A = 0 0 2 0, then its eigenvalues are ₁ = 1,λ₂ = 2, and 13 0 0 = : 0. (TRUE FALSE) ii) A linear transformation is operation preserving because the same result occurs whether you perform the operations of addition and scalar multiplication before or after applying the linear transformation. ( TRUE FALSE) iii) A linear transformation that is one-to-one and onto is called an isomorphism. (TRUE FALSE) iv) If the standard matrix A for the linear transformation T: R³ → R³ is -1 0 01 A = 2 00, then T is invertible. (TRUE FALSE) 0 1 1. v) Let A, B, and C be square matrices of order n. If A is similar to B and B is similar to C, then A is similar to C. ( TRUE FALSE) 2) a) i) Find the matrix that produces the counterclockwise rotation of 30° about the z-axis. ii) Find the image of the vector (1,1,1) for the rotation described in i). b) Give a geometric description of the linear transformations defined by the following matrix product: A = [{ ³] = [ ] }] [13] c) If the image of the vector (2,3) is (9,2) under a series of linear transformations, find the corresponding linear transformation matrix product and give a geometric description of the linear transformations.

Page < 2 of 2 - ZOOM + 3) a) Find the characteristic equation of the matrix b) Find the eigenvalues of A. A = [ 12 13³] c) Find each eigenvector corresponding to each eigenvalue of A. 4) a) Find a nonsingular matrix P such that P-1 AP is a diagonal matrix for the following matrix: A = 1 2 -21 -2 5 -2 -6 6 -3] b) Verify that P-1AP is a diagonal matrix with the eigenvalues on the main diagonal. Page < 1 of 2 - ZOOM + 1) Answer the following questions by circling TRUE or FALSE (No explanation or work required). −1 0 01 i) If A = 0 0 2 0, then its eigenvalues are ₁ = 1,λ₂ = 2, and 13 0 0 = : 0. (TRUE FALSE) ii) A linear transformation is operation preserving because the same result occurs whether you perform the operations of addition and scalar multiplication before or after applying the linear transformation. ( TRUE FALSE) iii) A linear transformation that is one-to-one and onto is called an isomorphism. (TRUE FALSE) iv) If the standard matrix A for the linear transformation T: R³ → R³ is -1 0 01 A = 2 00, then T is invertible. (TRUE FALSE) 0 1 1. v) Let A, B, and C be square matrices of order n. If A is similar to B and B is similar to C, then A is similar to C. ( TRUE FALSE) 2) a) i) Find the matrix that produces the counterclockwise rotation of 30° about the z-axis. ii) Find the image of the vector (1,1,1) for the rotation described in i). b) Give a geometric description of the linear transformations defined by the following matrix product: A = [{ ³] = [ ] }] [13] c) If the image of the vector (2,3) is (9,2) under a series of linear transformations, find the corresponding linear transformation matrix product and give a geometric description of the linear transformations.

Elementary Linear Algebra (MindTap Course List)
Elementary Linear Algebra (MindTap Course List)
8th Edition
ISBN:9781305658004
Author:Ron Larson
Publisher:Ron Larson
Chapter7: Eigenvalues And Eigenvectors
Section7.CR: Review Exercises
Problem 16CR
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Page < 2 of 2 - ZOOM + 3) a) Find the characteristic equation of the matrix b) Find the eigenvalues of A. A = [ 12 13³] c) Find each eigenvector corresponding to each eigenvalue of A. 4) a) Find a nonsingular matrix P such that P-1 AP is a diagonal matrix for the following matrix: A = 1 2 -21 -2 5 -2 -6 6 -3] b) Verify that P-1AP is a diagonal matrix with the eigenvalues on the main diagonal.
Transcribed Image Text:Page < 2 of 2 - ZOOM + 3) a) Find the characteristic equation of the matrix b) Find the eigenvalues of A. A = [ 12 13³] c) Find each eigenvector corresponding to each eigenvalue of A. 4) a) Find a nonsingular matrix P such that P-1 AP is a diagonal matrix for the following matrix: A = 1 2 -21 -2 5 -2 -6 6 -3] b) Verify that P-1AP is a diagonal matrix with the eigenvalues on the main diagonal.
Page < 1 of 2 - ZOOM + 1) Answer the following questions by circling TRUE or FALSE (No explanation or work required). −1 0 01 i) If A = 0 0 2 0, then its eigenvalues are ₁ = 1,λ₂ = 2, and 13 0 0 = : 0. (TRUE FALSE) ii) A linear transformation is operation preserving because the same result occurs whether you perform the operations of addition and scalar multiplication before or after applying the linear transformation. ( TRUE FALSE) iii) A linear transformation that is one-to-one and onto is called an isomorphism. (TRUE FALSE) iv) If the standard matrix A for the linear transformation T: R³ → R³ is -1 0 01 A = 2 00, then T is invertible. (TRUE FALSE) 0 1 1. v) Let A, B, and C be square matrices of order n. If A is similar to B and B is similar to C, then A is similar to C. ( TRUE FALSE) 2) a) i) Find the matrix that produces the counterclockwise rotation of 30° about the z-axis. ii) Find the image of the vector (1,1,1) for the rotation described in i). b) Give a geometric description of the linear transformations defined by the following matrix product: A = [{ ³] = [ ] }] [13] c) If the image of the vector (2,3) is (9,2) under a series of linear transformations, find the corresponding linear transformation matrix product and give a geometric description of the linear transformations.
Transcribed Image Text:Page < 1 of 2 - ZOOM + 1) Answer the following questions by circling TRUE or FALSE (No explanation or work required). −1 0 01 i) If A = 0 0 2 0, then its eigenvalues are ₁ = 1,λ₂ = 2, and 13 0 0 = : 0. (TRUE FALSE) ii) A linear transformation is operation preserving because the same result occurs whether you perform the operations of addition and scalar multiplication before or after applying the linear transformation. ( TRUE FALSE) iii) A linear transformation that is one-to-one and onto is called an isomorphism. (TRUE FALSE) iv) If the standard matrix A for the linear transformation T: R³ → R³ is -1 0 01 A = 2 00, then T is invertible. (TRUE FALSE) 0 1 1. v) Let A, B, and C be square matrices of order n. If A is similar to B and B is similar to C, then A is similar to C. ( TRUE FALSE) 2) a) i) Find the matrix that produces the counterclockwise rotation of 30° about the z-axis. ii) Find the image of the vector (1,1,1) for the rotation described in i). b) Give a geometric description of the linear transformations defined by the following matrix product: A = [{ ³] = [ ] }] [13] c) If the image of the vector (2,3) is (9,2) under a series of linear transformations, find the corresponding linear transformation matrix product and give a geometric description of the linear transformations.
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