MTH501 Midterm Past Paper 2

MIDTERM EXAMINATION(Solution File) · SEMESTER SPRING 2004 · MTH501- Linear Algebra · Instructions

1. The duration of this examination is 90 minutes.
2. This examination is closed book, closed notes, closed neighbors.
3. Answer all questions.
4. All the questions are descriptive in that paper, you can't answer yes or no only and if in certain questions your answer is yes or no then you have to justify your answer.
5. There is no mistake in the question paper and if you feel that there is something wrong with the question then made the best assumptions which you think and then give the answer.

Q1. Marks: 5

Determine the value(s) of h such that the matrix 1

h

is augmented matrix of a consistent system.

Solution: · 2 By · R · R · h · h

x1 + 4 x2 = -2 (-12 + h) x2 = 0 Therefore, the above system of equations is consistent for all h.

Q2. Marks: 5

Let

u

and

A

. Is u in the subset of R3 spanned by the columns of A? Why or why not?

Solution:

Consider u = a1 c1 + a2 c2 + a3 c3

It implies that

4a · 3a · 5a · a · a · a · 2a · 4a · 3a · 5a · a · a · (1) · a · 2a

Now

4(0 · 2) · 1( 3 · 5)

Therefore, the system (1) is inconsistent. Hence u is not in the subset of R3 spanned by the columns of A.

Q3. Assume that T is a linear transformation. Find the standard matrix of T.

T: R2  R4, T(e1) = (1, 2, 0, 5) and T(e2) = (3, -6, 1, 0), where e1 and e2 are the columns of 2 x 2 identity matrix.

Solution:

The standard matrix is:

A · T( · T( · e · e

Q4. Find the entries in the second row of AB, where

A · , B · Solution:

Q5. Solve the equation Ax = b by using the LU factorization given for A.

A · ,b · A · Solution:

Consider

Ly · b · Ux · y · y · Ly · b · y · y · 2,y · 2,y · y · x · Ux · y · x · x · 1/ 4,x · 2,x · x

Q6. Determine the matrix

is strictly diagonally dominant.

Solution:

Consider

| 9 | | 5| · | 2 | · | 8 | | 5| · | 1| · | 4 | | 2 | · |1|

Therefore, the given matrix is strictly diagonally dominant.

Q7. Let S be the parallelogram determined by the vectors

b

and

b

, and let

A

. Compute the area of the image of S under the mapping x  Ax.

Solution:

The area of S is

det

, and det A = 2. Therefore, by definition, the area of image of S under

the mapping x  Ax is |det A|.{area of S}= 2 . 14 = 28

Q8. Let

A

and

w

. Determine if w is in Col A?

Solution:

Consider w = a1 c1 + a2 c2 + a3 c3

The Augmented matrix is:

a1 + a3 = -1/2