## Answered) Assignment NO. 3 week9-week11 Student Full Name:___________________________________ . Student ID:__________________________________________ . CRN...

Assignment NO. 3 week9-week11
Student Full Name:___________________________________ .
Student ID:__________________________________________ .
CRN No:____________________________________________ .
Branch: _____________________________________________. Linear Algebra
(Math-251) Total Points
True/False ____/6 MCQ ____/6 Short Answer ____/18 Total ____/30 1|7Page Due date: 05 May 2017 Max Marks: 30 Section-I
State whether the following statements are true or false: [6X1=6] 1. Matrix is diagonalizable if it is similar to a diagonal matrix B;
that is, there exists an invertible matrix and = âˆ’1 .
(1)â€¦â€¦â€¦â€¦..
1 0
2. The eigenvalues of the matrix A =[âˆ’2 3
0 4
7. 0
0 ], are 1, 3, and âˆ’7
(2)â€¦â€¦â€¦â€¦.. 3. A square complex matrix is called Unitary if its conjugate
transpose equal to the matrix . (3)â€¦â€¦â€¦â€¦.. 4. The inner product of a nonzero vector with itself (âŒ© u, u âŒª ) is
always a positive real number. (4)â€¦â€¦â€¦â€¦.. 5. If = (1,2, âˆ’3,4) and = (âˆ’2,1,4,3) then âŒ©, âŒª = 0 , where
âŒ© , âŒª denotes the Euclidean inner product.
4
6. The matrix A = [2 âˆ’ 2 (5)â€¦â€¦â€¦â€¦.. 2âˆ’
2
3
âˆ’3 âˆ’ 6 ] is Hermitian.
6 âˆ’ 3
2
(6)â€¦â€¦â€¦â€¦.. 2|7Page Section-II
For Each Question, Choose the Correct Answer from the MultipleChoice:
[6X1=6]
1. Which of the following sets of vectors are orthogonal with
respect to the inner product, defined by âŒ© u, v âŒª = 21 1 +
52 2 on R2:
a. (2,5), (8,-3)
b. (3,4), (2,6)
c. (5,3), (3,-2)
d. (1,5), (3,-2)
2. The values of for which = (, 7, 2), and = (, âˆ’, 5) are
orthogonal in R3 with respect to Inner Product âŒ© u, v âŒª = 1 1 +
2 2 + 3 3 , are
a. 2, -5
b. -2, 5
c. -2,- 5
d. 2, 5
3. If 0 is an eigenvalue of a square matrix A then A is:
a. an Identity matrix.
b. invertible.
c. not invertible.
d. None
4 + 2i 1 + 8
4. If V = [ 2 âˆ’ 2i
1 âˆ’ 3i] , a complex matrix, then âˆ— is
8
6 âˆ’ 7
âˆ’5
4 âˆ’ 2i 1 âˆ’ 8
a. [ 2 + âˆ’2i
1 + 3i]
8
6 âˆ’ 7
âˆ’5
4 âˆ’ 2i 2 + 8
b. [ 1 âˆ’ âˆ’2i
âˆ’6i âˆ’ 7]
8
1 + 3
âˆ’5
4 âˆ’ 2i 2 âˆ’ 8
c. [ 1 + âˆ’2i
âˆ’7 âˆ’ 6i]
8
1 + 3
âˆ’5
4 + 2i 1 + 8
d. [ 2 âˆ’ 2i
1 âˆ’ 3i].
8
6 âˆ’ 7
âˆ’5
3|7Page 5
5. The characteristic equation of the matrix A = [
4
a. 2 âˆ’ 2 âˆ’ 19 = 0
b. 2 + 2 âˆ’ 19 = 0
c. 2 âˆ’ 2 + 19 = 0
d. 2 + 2 + 19 = 0 1 is
âˆ’3 6. Cosine value of the angle between vectors u and v is 1 and
|||| = 3, |||| = 5, then âŒ© u, v âŒª =
a. 10
b. 15
c. âˆš15
d. âˆš10
1 2 3 4 5 6 Section-III
Attempt all the questions: [6X3=18] 1. Find the complex dot product âˆ™ , such that = (5, âˆ’2, 2 +
, 4), = (3, 1 + 2, 1,3).
Solution: 4|7Page 2. Find all least square solutions and the error vector of the linear
system:
1 âˆ’ 2 = 2
1 + 2 = 4
21 + 2 = 8
Solution: 3. Show that matrix A = 1 âˆ’2 2 3
2 3
âˆ’1 3
âˆ’2 3
2 3
2 3
1 3 3 [3
Solution: 5|7Page is orthogonal and find A-1 4. Find the eigenvalues and the eigenvectors of the matrix
7 âˆ’4
=[
5 âˆ’2
Solution: 5. Show that matrix + is Hermitian , where =
0
0 3
1
âˆ’ 2
[ âˆ’2 0 ] , = [ 0
âˆ’1 1] .
2âˆ’ 0
1
âˆ’2 1 1 Solution: 6|7Page 6. Verify the property âŒ© u + v, w âŒª = âŒ© u, wâŒª + âŒ© v, w âŒª, If the inner
product on 2 , defined by âŒ© u, v âŒª = 21 1 + 32 2 .
Solution: 7|7Page

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