Algebra Questionnaire

Math 3330-002 Test 3 Version A, 05/14/2020 (5.1, 5.2, 6.1, 6.2, 7.2, 7.3)Last Name:
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2 
3 
  and  4k  are perpendicular.
1. Find all real value(s) of k such that k
 
 
 −1
0 
(A) They are perpendicular if k = 0.
(C) They are perpendicular if k = −3.
(B) They are not perpendicular for any values of k.
(D) They are perpendicular if k ≠ −2.
(E) None of the above
2. Let A be a 2 by 2 matrix. Its first row is [1 1], its second row is [1 2]. Find det A. Is A invertible?
(A)
det A = 2, A is invertible
(B) det A = 1, A is invertible
(D)
det A = 1, A is not invertible
(E) none of the above
(C) det A = 3, A is invertible
0 
2
3 
.
 −4
3. Perform the Gram-Schmidt process to the sequence of vectors: v1 =   and v2 = 
0 
1 
1 
0 
1 
2
(A) u1 =   , u2 =  
 −1
 , u2 =
0 
(D) u1 = 
1 
0 
0 
1 
 
1
1
1 
2
(C) u1 =   , u2 =  
(B) u1 =   , u2 =  
(E) None of the above
4. Lt A be a 4 by 4 matrix. Its first row is [1 0 0 2], its second row is [0 3 5 4], its third row is [0 0 7 3], its fourth
row is [0 0 0 3]. Find det (A).
(A) det A = 63
(B) det A = 72
(C) det A = 12
(D) det A = 96
(E) None of the above
5. Use the determinant to find out values for which values of k the matrix A is invertible. A is a 3×3 matrix whose first
row is [3 0 k], the second row is [1 0 1], the third row is [0 5 2].
(A) k = 1
(B) k = 2
(C) k ≠ 3
(D) k ≠ 2
(E) None of the above
6. Let matrix B be obtained from matrix A by three row swaps, and C be obtained from A by dividing a row of A by 5.
(A) det B = – det A, det C = (det A)/5
(B) det B = det A, det C = (det A)/3
(C) det B = – det A, det C = (det A)/2
(D) det B = det A, det C = (det A)/5
(E) None of the above
1 
2]v,
where v is any vector in R2. Let v1 =   and
1
0 
−1
7. Define a linear transformation T from R 2 to R2 by T(v) = [
1
0 
v2 =   form a basis. Let B be the matrix of T on this basis. Find B and det B.
1
 
(A) B =[
−1 1]
, det B = 1
3 0
−1 2]
(D) B =[
, det B = – 3
1 1
−1 1]
(B) B =[
, det B = – 3
1 2
−1 3]
(C) B =[
, det B = – 3
1 0
(E) None of the above
8. Find the derivative 𝑓 ′ (𝑡) of the function 𝑓(𝑡) = det A. A is a 4 by 4 matrix. Its first row is [1 0 1 5], its second row
is [2 0 2 0], its third row is [𝑡 5 3 8], its fourth row is [3 0 0 2]. Note: There is a 𝒕 in the matrix.
(A) 𝑓 ′ (𝑡) = 0
(B) 𝑓 ′ (𝑡) = 14
(C) 𝑓 ′ (𝑡) = 10
(D) 𝑓 ′ (𝑡) = 12
(E) None of the above
9. Use Gaussian elimination to find det A. A is a 4 by 4 matrix. Its first row is [2 1 0 2], its second row is
[−1 2 0 1], its third row is [1 1 0 1], its fourth row is [0 0 2 1].
(A) det A = 8
(B) det A = −4
(C) det A = 4
(D) det A = −8
(E) None of the above
2
 
10. Let v1 = 0 , v2 =
 
 0 
3 
4 , v =
  3
 0 
0 
0  . Perform the Gram-Schmidt process on the sequence of vectors.
 
3 
1 
 
(A) u1 = 0 , u2 =
 
0 
0 
0  , u =
  3
1 
0 
 −1 .
 
 0 
1 
 
(B) u1 = 0 , u2 =
 
0 
0 
0  , u =
  3
 −1
1 
 
(C) u1 = 0 , u2 =
 
0 
0 
1  , u =
  3
0 
0 
0  .
 
1 
1 
 
(D) u1 = 0 , u2 =
 
0 
0 
0  , u =
  3
1 
0 
1  .
 
0 
0 
1  .
 
0 
(E) None of the above
−1 2].
Determine their algebraic
−1 2
11. Find all distinct real eigenvalues and the associated eigenvectors for matrix A =[
multiplicity (almu).
(A) λ = 0, almu (0) = 1
(B) λ1 = 0, almu (0) = 2; λ2 = 1, almu (1) = 0
(C) λ = −1, almu (−1) = 1
(D) λ1 = 0, almu (0) = 1; λ2 = 1, almu (1) = 1
(E) None of the above
12. Let A be a 3 by 3 matrix. Its first row is [1 0 1], its second row is [0 2 1], its third row is [0 0 2]. Find all distinct real eigen
values of A. Then find a basis of each eigenspace. Determine whether A is diagonalizable.
0
1
(A) λ1 = 1, λ2 = 2; a basis of E1 is ( [1]); a basis of E2 is [0]; A is not diagonalizable.
0
0
0
0
0
(B) λ1 = −1, λ2 = 2; a basis of E−1 is ( [−1], [1]); a basis of E2 is[1]; A is diagonalizable.
1
1
0
1
0
(C) λ1 = 1, λ2 = 2; a basis of E1 is ([0]); a basis of E2 is ([1]); A is not diagonalizable.
0
0
0
0
0
(D) λ1 = −1, λ2 = 1; a basis of E−1 is ( [1] , [−1]); a basis of E1 is [1 ]; A is diagonalizable.
1
1
1
(E) None of the above