1. The system of equations
(x + 6y y = 1 = -12
is equivalent to which system of equations?
(A) (2xx — 52yy == 7-4 (B) (x x + 7 + y y = = 3-11 (C) (x x -+ 52y y = 4 = 7
(D) (2x x -+ 52yy == 7-4 (E) (2x x + 5 + 2y y = = 7-4
2. The linear system of equations
(3 6x x – – 2 4y y = 4 = 8
has
(A) No solution
(B) A unique solution
(C) An infinite number of solutions
(D) Exactly two solutions
(E) A trivial solution
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Math 1130 Linear Algebra
Math*1160 Take-Home Final (40 questions) April 24, 2020 (4pm)
3. If A = 0 @-1 2 1 5 1 31 A and B = -3 4 5 1 0 2, then what is (2A – 3BT )T ?
(A) 11 10 17 1 6 16 (B) -7 6 4 7 -14 -13
(C) -8 9 11 3 -11 -7 (D) 0 @- –14 6 13 4 7 71 A
(E) The expression cannot be calculated
4. If the numbers x; y; z; w 2 R satisfy the following matrix equation:
x2z z + 1 2 -yw = 24 2 x 1 :
Then
(A) (x; y; z; w) = (-1; 1=2; 4; 2)
(B) There is no solution
(C) (x; y; z; w) = (-1; 1=2; 4; 6)
(D) (x; y; z; w) = (1; 1=2; 4; 2)
(E) (x; y; z; w) = (1; 1=2; 4; 6)
3
5. Let the matrices A, B and C have dimensions m × r, r × n and n × k respectively. Then
the dimension of the product ABC is
(A) m × n
(B) m × k
(C) Not defined
(D) r × n
(E) k × m
6. The value(s) of k 2 R such that (kA)T (kA) + 1 = 0, where A = 0 @-2 021 A is
(A) -1=8
(B) Not a real number
(C) +1=8
(D) ±1=p3
(E) ±1=p8
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7. Let A; B and C be matrices. Then (BT AT CT )T =
(A) BAC
(B) BT AT CT
(C) CAB
(D) CT AT BT
(E) ABC
8. Consider the linear system AT x = b, where
A-1 = 2 3 4 -1 ; b = -31 :
Then the solution vector x is
(A) 15 1==11 11 (B) 7 5 (C) 15 1
(D) -37 (E) 15 1==11 11
5
9. Calculate (cA)-1 when c = 2 and A-1 = -2 3 1 0.
(A) -14 3 =2 0 =2 (B) -4 6 2 0 (C) -11==3 1 2 0 =6
(D) 4-=2 0 3 2=3 (E) -11 3 =2 0 =2
10. The matrix
0@
0 0 1 1 5
0 0 0 0 1
0 0 0 0 0
1A
is
(A) None of the other options apply
(B) Not in row echelon form
(C) In reduced row echelon form
(D) In row echelon form, but not reduced row echelon form
(E) In reduced row echelon form, but not in row echelon form
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11. The reduced row echelon form of
0BB@
1 2
1 3
0 1
1 0
1CCA
is
(A)
0BB@
1 0
0 0
0 0
0 1
1CCA
(B)
0BB@
1 1
0 0
0 0
0 0
1CCA
(C)
0BB@
1 0
2 0
0 0
0 0
1CCA
(D)
0BB@
1 0
0 1
0 0
0 0
1CCA
(E)
0BB@
1 2
0 1
0 0
0 0
1CCA
12. The linear system of equations
(-2x x1 1 -+ 53x x2 2 + x3 = 1 = -2
written as a single matrix equation (in its simplest form) is
(A) -21 5 0 -3 1 0 @x x x1 2 31 A = -12 (B) -21 5 0 -3 1 -12
(C) 0 @-21 0 3 5 -11 A 0 @x x x1 2 31 A = -12 (D) 0 @-2 0 0 0 1 5 0 -3 11 A 0 @x x x1 2 31 A = 0 @-1 021 A
(E) 0 @-21 0 3 5 -11 A x x1 2 = 0 @-1 021 A
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Math*1160 Take-Home Final (40 questions) April 24, 2020 (4pm)
13. The solution of the system
(-x1 + 2x x2 2 -+ x x3 3 = = – -2 1
is
(A) Does not exist
(B) f(3α; α – 1; α) j α 2 Rg
(C) f(3α; α – 1; 3α + 4) j α 2 Rg
(D) (x1; x2; x3) = (3; 0; 1)
(E) f(α; α – 1; 3α) j α 2 Rg
14. Let A be an n × n matrix. Which statement below is equivalent to the statement:
The non-homogeneous system Ax = b has a unique solution for every right
hand side vector b” ?
(A) jAj = 0
(B) The homogeneous system Ax = 0 has only the trivial solution
(C) The reduced row echelon form of A has at least one row of zeros
(D) The matrix A is upper triangular
(E) A is singular
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Math*1160 Take-Home Final (40 questions) April 24, 2020 (4pm)
15. If jAj = a b
c d = 5, then jBj = 5ac + 5 + b b d 5d is equal to
(A) 1
(B) -25
(C) 25
(D) -1
(E) 3
16. Let A and B be square matrices such that jAj = 2 and jBj = 3. Then jAT B-1j is equal to
(A) 2=3
(B) 6
(C) 3=2
(D) -6
(E) -3=2
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17. If A = 0 @1 3 0 0 5 0 0 6 -81 A then the homogeneous system of equation Ax = 0 has:
(A) Less than 3 solutions
(B) No solution
(C) A nontrivial solution
(D) Exactly 3 solutions
(E) Only the trivial solution
18. If A = 0 @2×2 4 6 x2 24xx2 36xx21 A, where x 2 R, then the cofactor expansion along the 1st row
yields jAj =
(A) 24×3
(B) 48×3 – 24×2 + 12x – 6
(C) 48×3
(D) 0
(E) 12
10
Math*1160 Take-Home Final (40 questions) April 24, 2020 (4pm)
19. Let A = 0 1 1 0. Then the adjoint of the matrix A, i.e. adj A, is given by
(A) 1 0 0 1 (B) 0 1 1 0 (C) -01 0 -1
(D) 01 0 -1 (E) -0 1 1 0
20. Let u = 3 2 , v = 1 4 and w = 1 2 . If xu + yv = w, (x; y 2 R),
then x + y =
(A) 1=5
(B) 2=5
(C) 1
(D) 4=5
(E) 3=5
11
21. The vectors OC ~ = u, CB ~ = v and AC ~ = w are illustrated below. OABC is a parallelogram.
From the diagram we see that w + v =
(A) u (B) 2u (C) u – v
(D) -u (E) u + v
22. The vectors OA ~ = v and OD ~ = u are represented below.
OACD and OABC are parallelograms. The vector OB ~ written in terms of the vectors u and
v is given by
(A) u + 2v (B) 2v – u (C) u + v
(D) u – v (E) v – u
12
23. Let V be a vector space with the operations of scalar multiplication and vector addition
⊕. Which rule below is NOT an axiom of a vector space, where u; v; w 2 V and c; d 2 R?:
(A) (u + v) + w = u + (v + w)
(B) (cd) u = c (d u)
(C) (c + d) u = c u ⊕ d u
(D) u ⊕ v = v ⊕ u
(E) c (u ⊕ v) = c u ⊕ c v
24. Let W be a subspace of a vector space V . Here ⊕ and represent vector addition and
scalar multiplication respectively. Which statement below is INCORRECT?
(A) k 2 R, u 2 W =) k u 2 W
(B) u; v 2 V =) u ⊕ v 2 W
(C) W is a subset of V
(D) W is a vector space with respect to the same operations of vector addition and scalar
multiplication as for V
(E) The zero vector 0 2 V belongs to W
13
25. Which statement below is INCORRECT?
(A) The set of vectors of the form 0 @a + 2 ab b1 A, a; b 2 R, is not a subspace of R3
(B) The set of points on the line x + 2y = 1 is not a subspace of R2
(C) W = f(x; 3x)T 2 R2 j x 2 Rg is a subspace of R2
(D) W = f(x1; x2) 2 R2 j x1 ≥ 0 and x2 ≥ 0g is not a subspace of R2
(E) W = f(x; 0; z) 2 R3 j x; z 2 Rg is a subspace of R3
26. Suppose the vectors v1; v2; : : : ; vk span a vector space V . Which statement below is INCORRECT?
(A) For any v 2 V there exists scalars a1; a2; : : : ; ak
such that a1v1 + a2v2 + · · · + akvk = v
(B) The vectors v1; v2; : : : ; vk can be viewed as the building blocks
of the space V
(C) V = fa1v1 + a2v2 + · · · + akvk j ai 2 Rg
(D) V = spanfv1; : : : ; vkg
(E) V ⊂ fa1v1 + a2v2 + · · · + akvk j ai 2 Rg (V is a proper subset)
14
Math*1160 Take-Home Final (40 questions) April 24, 2020 (4pm)
27. In the vector space M22 (set of all 2 × 2 matrices) the argument needed to determine if the
vector w = 1 0 2 1 belongs to the span of 3 1 2 -1 ; 2 1 1 0 yields the following
augmented matrix:
(A)
0BB@
3 2 0
-1 1 0
1 1 0
2 0 0
1CCA
(B)
0BB@
3 1 -1
1 2 0
2 1 2
1 0 1
1CCA
(C)
0BB@
3 1 -1
1 2 2
2 1 0
1 0 1
1CCA
(D) 3 1 2 1 1 0 -1 2 1 2 0 1 (E)
0BB@
3 2 1
-1 1 0
1 1 2
2 0 1
1CCA
28. In the vector space M22 (the set of all 2 × 2 matrices) the argument needed to determine if
the set of vectors 1 2 0 -1 ; 3 1 1 -2 is linearly independent leads to a homogeneous
system of linear equations in the form Ax = 0, where A =
(A) -1 2 3 1 1 0 1 -2 (B)
0BB@
1 3
-1 1
2 1
0 -2
1CCA
(C) 1 3 1 1 -1 2 0 -2
(D)
0BB@
2 1
0 -1
1 3
-2 1
1CCA
(E)
0BB@
1 -1
2 0
3 1
1 -2
1CCA
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Math*1160 Take-Home Final (40 questions) April 24, 2020 (4pm)
29. Suppose the vectors v1; v2; : : : ; vk in a vector space V are linearly independent. Which
statement does NOT necessarily follow?:
(A) None of the vectors v1; v2; : : : ; vk can be written as a linear combination of the other
vectors
(B) The span of the given vectors equals V
(C) The vectors are not linearly dependent
(D) There does not exist a linear relationship between the vectors v1; v2; : : : ; vk
(E)
kX i
=1
αivi = 0 =) αi = 0, i = 1; : : : k
30. The dimension of a vector space V is given by:
(A) The number of vectors in a basis for that space
(B) The number of vectors in a subset of V that is linearly independent
(C) The length times width of matrices in the space
(D) The number of vectors in a subset of V that spans the space
(E) The total number of vectors in the space
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31. Which statement concerning the set
S =
8<:
0@
2 0 0
1A
;
0@
0 –
3
0
1A
;
0@
0 0 4
1A
9=;
is CORRECT?
(A) S is a basis for R3
(B) S spans R3, but S is linearly dependent
(C) span S is a subset of (but NOT equal to) R3
(D) S is linearly independent, but does not span R3
(E) dim span S < 3
32. The equation of the straight line passing through the origin that corresponds to the null-space
of A = -31 2-=23 is:
(A) y = 2
3
x
(B) y = 3
2
x
(C) y = -6x
(D) y = -2
3
x
(E) y = -3
2
x
(Assume in the homogeneous system Ax = 0 we have x = (x; y)T .)
17
Math*1160 Take-Home Final (40 questions) April 24, 2020 (4pm)
33. The dimension of the null-space of A =
0BB@
1 2 0 0 3 0
0 0 1 0 4 0
0 0 0 1 5 0
0 0 0 0 0 1
1CCA
is
(A) 4
(B) 6
(C) 2
(D) 1
(E) 3
34. The rank of the matrix A =
0BB@
1 0 0 5 0
0 1 0 -2 0
0 0 1 3 0
0 0 0 0 1
1CCA
is
(A) 0
(B) 1
(C) 2
(D) 4
(E) 3
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35. Let A be an n × n matrix, and b and x be vectors of size n × 1. Which statement below
is NOT an equivalent condition for A to have full rank (rank A = n)? (0 is the n × 1 zero
vector)
(A) Ax = 0 =) x = 0
(B) Ax = b has a unique solution for every vector b
(C) A is nonsingular
(D) jAj = 0
(E) A is row reduceable to In
36. Let u = (1; 2; 3) and v = (-1; 0; 2), then the distance between u and v is given by
ku – vk =
(A) p8
(B) 2
(C) p5
(D) 3
(E) 1
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37. Consider an inner product (·; ·) defined on a vector space V . Let u; v; w 2 V and c 2 R.
Which statement below is NOT true for an inner product?
(A) (u; v) = (v; u)
(B) (u; u) ≥ 0 with equality iff u = 0
(C) (u + v; w) = (u; w) + (v; w)
(D) (cu; v) = c(u; v)
(E) An inner product is a map (·; ·) from V × V to Rn
38. Consider the standard inner product (u; v) := u · v, where u; v 2 R3. If
u =
0@
-2
-1
2
1A
and v = 0 @p2 331 A ;
then the Cauchy-Schwartz Inequality yields an upper bound on ju · vj to be
(A) 122
(B) 4p8
(C) 3p15
(D) 2 – p3
(E) 12
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39. The eigenvalues of the matrix 0 1 1 0 are +1 and -1. If α and β are arbitrary constants,
then the associated eigenvectors can be written as (respectively):
(A) α 0 1 and β 1 0 (B) α -11 and β 1 1 (C) α 1 2 and β -21
(D) α 1 1 and β -11 (E) α 1 0 and β -11
40. The matrix A = 1 0 2 3 has eigenvalues λ1 = 1 and λ2 = 3 with associated eigenvectors
x
1 = -11 and x2 = 0 1 respectively. The matrices P and D such that D = P -1AP
are given by
(A) D = 1 0 0 3 and P = -1 1 1 0 (B) D = 1 0 0 3 and P = -0 1 1 1
(C) D = 3 0 0 1 and P = -0 1 1 1 (D) D = 1 0 0 3 and P = 0 1 1 -1
(E) D = 3 0 0 1 and P = -1 1 1 0
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