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

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