Prepare for Linear Algebra Test 1

True or False and why?

§1.1 #55

1. A system of one linear equation in two variables is always consistent.
2. A system of two linear equations in three unknowns is always consistent.
3. If a linear system is consistent, then it has an infinite number of solutions .

§1.1 #56

4. A system of linear equations can have exactly two solutions.
5. Two systems of linear equations are equivalent if they have the same solution set.
6. A system of three linear equations in two unknowns is always inconsistent.

§1.2 #51

7. A 6 × 3 matrix has six rows.
8. Every matrix is row-equivalent to a matrix in row-echelon form.
9. If the row-echelon form of the augmented matrix of a system of linear equations contains
the row [10000], then the original system is inconsistent.
10. A homogeneous systems of four linear equations in six unknowns has an infinite number
of solutions.

§1.2 #52

11. A 4 × 7 matrix has four columns.
12. Every matrix has a unique reduced row -echelon form.
13. A homogeneous system of four linear equations in four unknowns is always consistent.
14. Multiplying a row of a matrix by a constant is one of the elementary row operations.

§2.1 #47

15. For the product of two matrices to be defined, the number of columns of the first matrix
must equal the number of rows of the second matrix.
16. The system Ax = b is consistent if and only if b can be ex pressed as a linear combination
where the coefficients of the linear combination are a solution to the system.

§2.1 #48

17. If A is an m × n matrix and B is an n × r matrix, then the product AB is an m × r
matrix.
18. The matrix equation Ax = b, where A is the coefficient matrix and x and b are column
matrices, can be used to represent a system of linear equations.

§2.2 #31

19. Matrix addition is commutative .
20. Matrix multiplication is associative.
21. The transpose of the product of two matrices equals the product of their transposes, that
is (AB)T = ATBT .
22. For any matrix C. the matrix CCT is symmetric.

§2.2 #32

23. Matrix multiplication is commutative.
24. Every matrix A has an additive inverse.
25. If the matrices A, B, and C satisfy AB = AC, then B = C.
26. The transpose of the sum of two matrices equals the sum of their transposes.

§2.3 #37

27. The inverse of a nonsingular matrix is unique.
28. If the matrices A, B, and C satisfy BA = CA, and A is invertible, then B = C.
29. The inverse of the product of two matrices is the product of their inverses, that is
(AB)-1 = A-1B-1.
30. If A can be row reduced to the identity matrix, then A is nonsingular.

§2.3 #38

31. The product of four invertible 7 × 7 matrices is invertible.
32. The transpose of the inverse of a nonsingular matrix is equal to the inverse of the transpose.
33. The matrix is invertible if ab - dc ≠ 0.
34. If A is a square matrix , then the system of linear equations Ax = b has a unique solution.

§3.1 #43

35. The de terminant of the 2 × 2 matrix A is .
36. The determinant of a matrix of order 1 is the entry of the matrix.
37. The ij-co factor of a square matrix A is the matrix defined by deleting the ith row and
the jth column of A.

§3.1 #44

38. To find the determinant of a triangular matrix, add the entries on the main diagonal.
39. The determinant of a matrix can be evaluated using expansion by cofactors in any row
or column.
40. When expanding by cofactors, you need not evaluate the cofactors of zero entries .

§3.2 #29

41. Interchanging two rows of a given matrix changes the sign of its determinant.
42. Multiplying a row by a nonzero constant results in the determinant of the matrix being
multiplied by the same nonzero constant.
43. If two rows of a square matrix are equal, then its determinant is equal to 0.

§3.2 #30

44. Adding a multiple of one row of a matrix to another row changes only the sign of its
determinant.
45. Two matrices are column-equivalent if one matrix can be obtained from the other by
elementary column operations.
46. If one row of a square matrix is a multiple of another row, then its determinant is equal
to 0.

§3.3 #47

47. If A is an n × n matrix and c is a nonzero scalar, then the determinant of the matrix cA
is given by nc · det(a).
48. If A is an invertible matrix, then the determinant of A-1 is equal to the reciprocal of the
determinant
of A.
49. If A is an invertible n × n matrix, then Ax = b has a unique solution for every b.

§3.3 #48

50. In general, the determinant of the sum of two matrices equals the sum of the determinants
of the matrices.
51. If A is a square matrix, then the determinant of A is equal to the determinant of the
transpose of A.
52. If the determinant of an n × n matrix A is nonzero, then Ax = 0 has only the trivial
solution.

§4.1 #35

53. Two vectors in are equal if and only if their corresponding components are equal.
54. For a nonzero scalar c and a vector v in , the vector cv is c times as long as v and has
the same direction as v

§4.1 #36

56. The zero vector 0 in is defined as the additive inverse of a vector.

§4.2 #31

57. The objects in a vector space are vectors.
58. The set of all integers with standard ope rations is a vector space.
59. The set of all pairs of real numbers of the form (x, y), where y ≥ 0, with standard
operations in is a vector space.

§4.2 #32

60. It is sufficient to show that a set is not a vector space by showing that just one axiom is
not satisfied.
61. The set of all first-degree polynomials with standard operations is a vector space.
62. The set of all pairs of real numbers of the form (0, y) with the standard operations on
is a vector space.

§4.3 #21

63. Every vector space V contains at least one subspace that is the zero subspace.
64. If V and W are both subspaces of a vector space U, then the intersection of V and W is
also a subspace.
65. If U, V , and W are vector spaces such that W is a subspace of V and U is a subspace of
V , then W = U.

§4.3 #22

66. Every vector space V contains two proper subspaces that are the zero subspace and itself.
67. If W is a subspace of , then W must contain the vector (0, 0).
68. If W and U are subspaces of a vector space V , then the union of W and U is a subspace
of V .

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