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Problem 6: Given A, B E Maxn(R), which one of the following statements is false: A) If λ is an eigenvalue of A with associated eigenvector v then v is an eigenvector associated with the eigenvalue A². B) The determinant of A is always the product of its pivots. C) AB is orthogonal if A and B are both orthogonal matrices. D) The determinant of AB BA is 0. E) det A=0 if and only if rref(A) # In. Problem 7: If A is a 5 x 7 matrix with rank 3, which one of the following statements is true: A) The column space of A has dimension 2. B) The row space of A has dimension 4. C) The null space of A has dimension 4. D) Az = 0 has only the trivial solution. L E) A-¹ can be written as a multiplication of elementary matrices.

Problem 6: Given A, B E Maxn(R), which one of the following statements is false: A) If λ is an eigenvalue of A with associated eigenvector v then v is an eigenvector associated with the eigenvalue A². B) The determinant of A is always the product of its pivots. C) AB is orthogonal if A and B are both orthogonal matrices. D) The determinant of AB BA is 0. E) det A=0 if and only if rref(A) # In. Problem 7: If A is a 5 x 7 matrix with rank 3, which one of the following statements is true: A) The column space of A has dimension 2. B) The row space of A has dimension 4. C) The null space of A has dimension 4. D) Az = 0 has only the trivial solution. L E) A-¹ can be written as a multiplication of elementary matrices.

Linear Algebra: A Modern Introduction
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter6: Vector Spaces
Section6.2: Linear Independence, Basis, And Dimension
Problem 5AEXP
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Problem 6: Given A, B E Maxn(R), which one of the following statements is false: A) If λ is an eigenvalue of A with associated eigenvector v then v is an eigenvector of A2, associated with the eigenvalue A². B) The determinant of A is always the product of its pivots. C) AB is orthogonal if A and B are both orthogonal matrices. D) The determinant of AB BA is 0. E) det A = 0 if and only if rref(A) # In. Problem 7: If A is a 5 x 7 matrix with rank 3, which one of the following statements is true: A) The column space of A has dimension 2. B) The row space of A has dimension 4. C) The null space of A has dimension 4. D) Az = 0 has only the trivial solution. E) A-¹ can be written as a multiplication of elementary matrices. Problem 8: Find the determinant of the matrix A) (k-1)(k+2) B) k C) k-1 D) k + 2 E) 0 k k k -1 -1 k 2 k 3 where k 0.
Transcribed Image Text:Problem 6: Given A, B E Maxn(R), which one of the following statements is false: A) If λ is an eigenvalue of A with associated eigenvector v then v is an eigenvector of A2, associated with the eigenvalue A². B) The determinant of A is always the product of its pivots. C) AB is orthogonal if A and B are both orthogonal matrices. D) The determinant of AB BA is 0. E) det A = 0 if and only if rref(A) # In. Problem 7: If A is a 5 x 7 matrix with rank 3, which one of the following statements is true: A) The column space of A has dimension 2. B) The row space of A has dimension 4. C) The null space of A has dimension 4. D) Az = 0 has only the trivial solution. E) A-¹ can be written as a multiplication of elementary matrices. Problem 8: Find the determinant of the matrix A) (k-1)(k+2) B) k C) k-1 D) k + 2 E) 0 k k k -1 -1 k 2 k 3 where k 0.
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