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[answered] Ministry of Higher Education Kingdom of Saudi Arabia CSTS S

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Ministry of Higher Education

Kingdom of Saudi Arabia CSTS

SEU, KSA Linear Algebra (Math 251)

Level IV, Assignment 3

(Fall, 2016) 1. State whether the following statements are true or false: [6] (a) If 2, 3 and 4 are eigen values of a matrix A, then det(A) = 9.

(a)

(b) (1,-1,2) is the real part of the complex vector (1 + i, ?1 + i, i, 2).

(b)

(c) The inner product of a nonzero vector with itself is always a positive real number.

(c)

(d) If u = (1, ?1), v = (?2, 2) and k = 4, then < ku, v >= 16.

(d)

(e) If determinant of a matrix is 1 or -1, then the matrix is orthogonal.

(e)

(f) The rows and columns of an orthogonal matrix are orthonormal.

(f) Page 1 of 3 Last Date: 14 Dec, 2016 Math 251 Department of Mathematics 2. Select one of the alternatives from the following questions as your answer.

3 1

(a) The characteristic equation of the matrix A =

is

2 4

A. ?2 ? 7? ? 10 = 0

B. ?2 + 7? ? 10 = 0

C. ?2 ? 7? + 10 = 0

D. ?2 + 7? + 10 = 0 1 ?1 2

1 , are

(b) The eigenvalues of the matrix A3 , where A = 0 4

0 0 ?3

A. {1, 4, ?3}

B. {1, 12, ?9}

C. {1, 64, 27}

D. {1, 64, ?27}

(c) Which of the following sets of vectors are orthogonal with respect to the Euclidean

inner product on R2 :

A.

B.

C.

D. (1,2), (-2,1)

(3,4),(2,6)

(6,9),(5,2)

(0,4), (0,6) (d) If angle between vectors u and v is zero such that kuk = 4, kvk = 6, then < u, v >=

A. 10

B. 24

?

C. 24

?

D. 10

(e) If 3x21 +2x22 ?4x23 ?2x1 x2 +6x1 x3 ?4x2 x3 be the quadratic form, then the associated

symmetric matrix will be 3 1

3

A. 1 2 ?2 3 ?2 ?4 3 ?1 ?3

B. ?1 2 ?2 ?3 ?2 ?4 3 ?1 3

2 C. ?1 2

3

2 ?4 Page 2 of 3 Last Date: 14 Dec, 2016 [6] Math 251 Department of Mathematics 3 ?1 3

D. ?1 2 ?2 3 ?2 ?4 3

i + 2 2 + 6i

?1 2i ? 1 is Hermitian?

(f) For which value of a and b, the matrix a

2 ? 6i

b

1

A. a = i ? 2, b = ?2i ? 1

B. a = ?i + 2, b = 2i + 1

C. a = ?i + 2, b = ?2i ? 1

D. a = i ? 2, b = 2i + 1

5 ?2

3. Find all the eigenvalues and the corresponding eigenvectors of the matrix A =

.

3 0

1 + 3i

2

4. For the matrix A =

Find A, Re(A), Im(A), T r(A) and det(A).

3+i 4?i

9 4

2 ?2

5. Find the value of k, for which the matrices U =

and V =

are

2 6

k 4

orthogonal in the vector space M2?2 with usual inner product on M2?2 .

6. Find 1 2

2 the least squares

solution

of the system of linear equation AX = B, where A = ?3

?2

1 , B = 2 .

1

0 1

1?i ?

? ? 3 7. Show that the matrix A = 1 +3i is unitary.

1

?

?

3

3 2 3

6 7 7

7 3

6

2 is orthogonal and find A?1 .

8. Show that matrix A = ? 7 6 27 73 ?

7 7

7 Page 3 of 3 End of Assignment. [4]

[2]

[2] [4] [3] [3]

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