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(12) (7 marks) Let the distinct eigenvalues of a square matrix A be denoted by 11,...
(7 marks) Let the distinct eigenvalues of a square matrix A be denoted by 11, ...,...
(7 marks) Let the distinct eigenvalues of a square matrix A be denoted by 11, ..., lk. Suppose the corresponding algebraic multiplicities are mi, ..., Mk and that A is similar to an upper-triangular matrix. Show that k k tr(A) midi and det(A) = II (4;)mi i=1 i=1
8. Let A be an nxn matrix with distinct n eigenvalues X1, 2... (a) What is...
8. Let A be an nxn matrix with distinct n eigenvalues X1, 2... (a) What is the determinant of A. (b) If a 2 x 2 matrix satisfies tr(AP) = 5, tr(A) = 3, then find det(A). (The trace of a square matrix A, denoted by tr(A), is the sum of the elements on the main diagonal of A.
(1 point) Suppose a 3 x 3 matrix A has only two distinct eigenvalues. Suppose that...
(1 point) Suppose a 3 x 3 matrix A has only two distinct eigenvalues. Suppose that tr(A) = 1 and det(A) = 63. Find the eigenvalues of A with their algebraic multiplicities. The smaller eigenvalue has multiplicity and the larger eigenvalue has multiplicity
Question 1 1 pts Cis a 3x3 matrix with exactly two distinct eigenvalues, 11 and 12....
Question 1 1 pts Cis a 3x3 matrix with exactly two distinct eigenvalues, 11 and 12. Which of the following are possibilities for the algebraic and geometric multiplicities of l, and 12 as eigenvalues of C? (select ALL that apply) It is possible that 11 has algebraic multiplicity 2 and geometric multiplicity 2, and X2 has algebraic multiplicity 1 and geometric multiplicityo. It is possible that X has algebraic multiplicity 2 and geometric multiplicity 1, and 12 has algebraic multiplicity...
4. (a) (6 marks) Let A be a square matrix with eigenvector v, and corresponding eigenvalue...
4. (a) (6 marks) Let A be a square matrix with eigenvector v, and corresponding eigenvalue 1. Let c be a scalar. Show that A-ch has eigenvector v, and corresponding eigenvalue X-c. (b) (8 marks) Let A = (33) i. Find the eigenvalues of A. ii. For one of the eigenvalues you have found, calculate the corresponding eigenvector. iii. Make use of part (a) to determine an eigenvalue and a corresponding eigenvector 2 2 of 5 - 1
True or false. Please justify why true or why false also (I) A square matrix with...
True or false. Please justify
why true or why false also
(I) A square matrix with the characteristic polynomial 14 – 413 +212 – +3 is invertible. [ 23] (II) Matrix in Z5 has two distinct eigenvalues. 1 4 (III) Similar matrices have the same eigenspaces for the corresponding eigenvalues. (IV) There exists a matrix A with eigenvalue 5 whose algebraic multiplicity is 2 and geo- metric multiplicity is 3. (V) Two diagonal matrices D1 and D2 are similar if...
(1) (5 marks) True or False? Justify your answer. Answers without correct justification will receive no...
(1) (5 marks) True or False? Justify your answer. Answers without correct justification will receive no credit. (1) A square matrix with the characteristic polynomial X - 413 +212 - +3 is invertible. [23] (II) Matrix in Zs has two distinct eigenvalues. (III) Similar matrices have the same eigenspaces for the corresponding eigenvalues. (IV) There exists a matrix A with eigenvalue 5 whose algebraic multiplicity is 2 and geo- metric multiplicity is 3. (V) Two diagonal matrices Dand D2 are...
(1 point) Find the three distinct real eigenvalues of the upper-triangular matrix B= 5-7 0 0...
(1 point) Find the three distinct real eigenvalues of the upper-triangular matrix B= 5-7 0 0 7 -1 0 -97 -4 . 4 The eigenvalues are [Note: If there is more than one answer, separate them by commas. E.g. 1,2]
Let A be an n x n matrix. Then we know the following facts: 1) IfR"...
Let A be an n x n matrix. Then we know the following facts: 1) IfR" has a basis of eigenvectors corresponding to the matrix A, then we can factor the matrix as A = PDP-1 2) If ) is an eigenvalue with algebraic multiplicity equal to k > 1, then the dimension of the A-eigenspace is less than or equal to k. Then if the n x n matrix A has n distinct eigenvalues it can always be factored...
6.3 (Adjugate matrix) a) Let A E GLn(K). Use Cramer's rule to show that A-1 =...
6.3 (Adjugate matrix) a) Let A E GLn(K). Use Cramer's rule to show that A-1 = data adj A without using Lemma 4.5.17. b) Let A Knxn be an upper triangular matrix (i.e. ajj = 0 (1 <j<i<n). Show that adj A is an upper triangular matrix. c) Let Znxn := {A € RNXN | aij € Z (i, j = 1, ..., n)}. Show that U := {A € Znxn | det(A) = 1} is a group with respect...
(7 marks) Let the distinct eigenvalues of a square matrix A be denoted by 11, ..., lk. Suppose the corresponding algebraic multiplicities are mi, ..., Mk and that A is similar to an upper-triangular matrix. Show that k k tr(A) midi and det(A) = II (4;)mi i=1 i=1
8. Let A be an nxn matrix with distinct n eigenvalues X1, 2... (a) What is the determinant of A. (b) If a 2 x 2 matrix satisfies tr(AP) = 5, tr(A) = 3, then find det(A). (The trace of a square matrix A, denoted by tr(A), is the sum of the elements on the main diagonal of A.
(1 point) Suppose a 3 x 3 matrix A has only two distinct eigenvalues. Suppose that tr(A) = 1 and det(A) = 63. Find the eigenvalues of A with their algebraic multiplicities. The smaller eigenvalue has multiplicity and the larger eigenvalue has multiplicity
Question 1 1 pts Cis a 3x3 matrix with exactly two distinct eigenvalues, 11 and 12. Which of the following are possibilities for the algebraic and geometric multiplicities of l, and 12 as eigenvalues of C? (select ALL that apply) It is possible that 11 has algebraic multiplicity 2 and geometric multiplicity 2, and X2 has algebraic multiplicity 1 and geometric multiplicityo. It is possible that X has algebraic multiplicity 2 and geometric multiplicity 1, and 12 has algebraic multiplicity...
4. (a) (6 marks) Let A be a square matrix with eigenvector v, and corresponding eigenvalue 1. Let c be a scalar. Show that A-ch has eigenvector v, and corresponding eigenvalue X-c. (b) (8 marks) Let A = (33) i. Find the eigenvalues of A. ii. For one of the eigenvalues you have found, calculate the corresponding eigenvector. iii. Make use of part (a) to determine an eigenvalue and a corresponding eigenvector 2 2 of 5 - 1
True or false. Please justify
why true or why false also
(I) A square matrix with the characteristic polynomial 14 – 413 +212 – +3 is invertible. [ 23] (II) Matrix in Z5 has two distinct eigenvalues. 1 4 (III) Similar matrices have the same eigenspaces for the corresponding eigenvalues. (IV) There exists a matrix A with eigenvalue 5 whose algebraic multiplicity is 2 and geo- metric multiplicity is 3. (V) Two diagonal matrices D1 and D2 are similar if...
(1) (5 marks) True or False? Justify your answer. Answers without correct justification will receive no credit. (1) A square matrix with the characteristic polynomial X - 413 +212 - +3 is invertible. [23] (II) Matrix in Zs has two distinct eigenvalues. (III) Similar matrices have the same eigenspaces for the corresponding eigenvalues. (IV) There exists a matrix A with eigenvalue 5 whose algebraic multiplicity is 2 and geo- metric multiplicity is 3. (V) Two diagonal matrices Dand D2 are...
(1 point) Find the three distinct real eigenvalues of the upper-triangular matrix B= 5-7 0 0 7 -1 0 -97 -4 . 4 The eigenvalues are [Note: If there is more than one answer, separate them by commas. E.g. 1,2]
Let A be an n x n matrix. Then we know the following facts: 1) IfR" has a basis of eigenvectors corresponding to the matrix A, then we can factor the matrix as A = PDP-1 2) If ) is an eigenvalue with algebraic multiplicity equal to k > 1, then the dimension of the A-eigenspace is less than or equal to k. Then if the n x n matrix A has n distinct eigenvalues it can always be factored...
6.3 (Adjugate matrix) a) Let A E GLn(K). Use Cramer's rule to show that A-1 = data adj A without using Lemma 4.5.17. b) Let A Knxn be an upper triangular matrix (i.e. ajj = 0 (1 <j<i<n). Show that adj A is an upper triangular matrix. c) Let Znxn := {A € RNXN | aij € Z (i, j = 1, ..., n)}. Show that U := {A € Znxn | det(A) = 1} is a group with respect...
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