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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 = −1 .
(1)…………..
1 0
2. The eigenvalues of the matrix A =[−2 3
0 4
7. 0
0 ], are 1, 3, and −7
(2)………….. 3. A square complex matrix is called Unitary if its conjugate
transpose equal to the matrix . (3)………….. 4. The inner product of a nonzero vector with itself (〈 u, u 〉 ) is
always a positive real number. (4)………….. 5. If = (1,2, −3,4) and = (−2,1,4,3) then 〈, 〉 = 0 , where
〈 , 〉 denotes the Euclidean inner product.
4
6. The matrix A = [2 − 2 (5)………….. 2−
2
3
−3 − 6 ] is Hermitian.
6 − 3
2
(6)………….. 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 〈 u, v 〉 = 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 = (, −, 5) are
orthogonal in R3 with respect to Inner Product 〈 u, v 〉 = 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 − 2i
1 − 3i] , a complex matrix, then ∗ is
8
6 − 7
−5
4 − 2i 1 − 8
a. [ 2 + −2i
1 + 3i]
8
6 − 7
−5
4 − 2i 2 + 8
b. [ 1 − −2i
−6i − 7]
8
1 + 3
−5
4 − 2i 2 − 8
c. [ 1 + −2i
−7 − 6i]
8
1 + 3
−5
4 + 2i 1 + 8
d. [ 2 − 2i
1 − 3i].
8
6 − 7
−5
3|7Page 5
5. The characteristic equation of the matrix A = [
4
a. 2 − 2 − 19 = 0
b. 2 + 2 − 19 = 0
c. 2 − 2 + 19 = 0
d. 2 + 2 + 19 = 0 1 is
−3 6. Cosine value of the angle between vectors u and v is 1 and
|||| = 3, |||| = 5, then 〈 u, v 〉 =
a. 10
b. 15
c. √15
d. √10
Answer:
1 2 3 4 5 6 Section-III
Attempt all the questions: [6X3=18] 1. Find the complex dot product ∙ , such that = (5, −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 − 2 = 2
1 + 2 = 4
21 + 2 = 8
Solution: 3. Show that matrix A = 1 −2 2 3
2 3
−1 3
−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 −4
=[
5 −2
Solution: 5. Show that matrix + is Hermitian , where =
0
0 3
1
− 2
[ −2 0 ] , = [ 0
−1 1] .
2− 0
1
−2 1 1 Solution: 6|7Page 6. Verify the property 〈 u + v, w 〉 = 〈 u, w〉 + 〈 v, w 〉, If the inner
product on 2 , defined by 〈 u, v 〉 = 21 1 + 32 2 .
Solution: 7|7Page

 


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