1. If a matrix A is row equivalent to B, then Col A = Col B. True or False 2. If

ID: 3143292 • Letter: 1

Question

1. If a matrix A is row equivalent to B, then Col A = Col B. True or False

2. If a matrix A is row equivalent to B, then Row A = Row B. True or False

3. If a matrix A is row equivalent to B, then Nul A = Nul B. True or False

4. If a matrix A is row equivalent to B, then rank A = rank B. True or False

5. If a matrix A is row equivalent to B, then det A = det B. True or False

6. For any matrix A, dim Col A = dim Row A. True or False

7. If A is an m × n matrix, then dim Col A + dim Nul A = n. True or False

8. If Ax = b is consistent, then b is in Col A. True or False

9. Col A is the set of all solutions of Ax = b. True or False 3

10. If B is an echelon form of a matrix A, then the pivot columns of B form a basis for Col A. True or False

11. Let A and B be n × n matrices. Then det AB = (det A)(det B). True or False

12. Let A and B be n×n matrices. Then det(A+ B) = det A+ det B. True or False

13. If the products are defined, then (AB) T = BTAT . True or False

14. If there is a linearly independent set {v1, v2, . . . , vp} in V , then dim V p. True or False

15. Vectors in Row A are orthogonal to vectors in Nul A. True or False

16. If a matrix A is row equivalent to B, then A and B have the same eigenvalues. True or False

17. If A is similar to B (i.e., A = P BP 1 ), then det A = det B. True or False

18. If A is similar to B (i.e., A = P BP 1 ), then A and B have the same eigenvalues. True or False

19. If an eigenvalue is 0, then the corresponding eigenvector is 0. True or False

20. If x is in W, then x projW x is zero. True or False

(According to Chegg policy only four subquestions will be answered. Please post the remaining in another question)

1. If a matrix A is row equivalent to B, then Col A = Col B. True or False

Answer: False. The column spaces need not be the same.

2. If a matrix A is row equivalent to B, then Row A = Row B. True or False

True as per definition.

3. If a matrix A is row equivalent to B, then Nul A = Nul B. True or False

True because the null space of a matrix is the orthogonal complement of the row space.

4. If a matrix A is row equivalent to B, then rank A = rank B. True or False

True because the rank of a matrix is equal to the dimension of the row space.

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