MATH 3321 Quiz 15 11/10/14

1. If the rank of the augmented matrix of a system of n linear equations in n unknownsis greater than the rank of the matrix of coefficients, then the matrix of coefficients isnonsingular.

(a) Always true

(b) Sometimes true

(c) Never true, i.e., false

(d) None of the above

2. If the matrix of coefficients of a homogeneous system of n linear equations in n unknownshas an inverse, then the system does not have infinitely many solutions.

(a) Always true

(b) Sometimes true

(c) Never true, i.e., false

(d) None of the above

3. If a system of n linear equations in n unknowns is inconsistent, then the reduced rowechelon form of the matrix of coefficients is not In.

(a) Always true

(b) Sometimes true

(c) Never true, i.e., false

(d) None of the above

4. If the matrix of coefficients of a homogeneous system of n linear equations in n unknownsis nonsingular, then the trivial solution is the only solution of the system.

(a) Always true

(b) Sometimes true

(c) Never true, i.e., false

(d) None of the above

5. The rank of the matrix of coefficients of a homogeneous system of m linear equations in n

unknowns is never less than the rank of the augmented matrix.

(a) Always true

(b) Sometimes true

(c) Never true, i.e., false

(d) None of the above

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6. If the reduced row echelon form of the matrix of coefficients of a system of n linear equationsin n unknowns is not the identity, then the system is inconsistent.

(a) Always true

(b) Sometimes true

(c) Never true, i.e., false

(d) None of the above

7. The value(s) of a for which that the vectors

v1 = (a, 0, 2), v2 = (0, 4, a), v3 = (−1, 2, a)

are linearly independent is(are):

(a) a = ±2

(b) a 6= ±2

(c) The vectors are linearly independent for all real numbers a.

(d) a = −4

(e) The vectors are linearly dependent for all real numbers a.

8. The value(s) of λ such that the rows of

−4 1 20 1 λ

λ 0 2

are linearly dependent is(are):

(a) λ 6= 4, −2

(b) λ = 2, −4

(c) λ 6= 4, 2

(d) λ = 4, −2

(e) The rows are linearly dependent for all real numbers λ.

9. The maximum number of independent vectors in the set

{v1 = ( 2, 0, −1 ), v2 = (−3, 1, 2 ), v3 = ( 8, −2, −5 ), v4 = (−9, 1, 5 )}

is:

(a) 4

(b) 3

(c) 2

(d) 1

(e) 0

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10. The value(s) of λ for which the matrix

1 − λ 2 03 −λ 04 2 4 − λ

is singular is(are):

(a) λ = 4, 3, −2

(b) λ = 4, −3

(c) λ = 4, −3, 1.

(d) λ = −4, −2, 3

(e) The matrix is singular for all real numbers λ.

11. The value(s) of a for which the homogeneous system of equations

6 3 + a −13 5 −4

a − 1 −4 6

x

y

z

=

000

has non-trivial solutions is(are):

(a) a = −5, 7

(b) a = −7, 7

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(c) a 6= 7, −10

7

(d) a = 7, −7

(e) The system has only the trivial solution for all values of a.

12. The value(s) of a such that the vectors

v1 = (5, 3, 0), v2 = (0, 1, a), v3 = (0, 2 − a, 1)

are linearly dependent is(are):

(a) a = 5, 1

(b) a 6= −1

(c) The vectors are linearly dependent for all real numbers a.

(d) a = 1

(e) a = 1, −2.

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