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Assignment 1

This document contains a 20-question mathematics assignment for engineering students. The questions cover topics such as vector operations, matrix operations including determinants and inverses, solving systems of linear equations using Gaussian elimination and Cramer's rule, and checking linear dependence of vectors. Students are instructed to show their work clearly, submit a single PDF with problems numbered, and name the file with their identifying information.

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Abbas Tufan
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49 views2 pages

Assignment 1

This document contains a 20-question mathematics assignment for engineering students. The questions cover topics such as vector operations, matrix operations including determinants and inverses, solving systems of linear equations using Gaussian elimination and Cramer's rule, and checking linear dependence of vectors. Students are instructed to show their work clearly, submit a single PDF with problems numbered, and name the file with their identifying information.

Uploaded by

Abbas Tufan
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
Available Formats
Download as PDF, TXT or read online on Scribd
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MIDDLE EAST TECHNICAL UNIVERSITY

ES202 – Mathematics for Engineers

Murat Büyük, PhD

Assignment-1

1) Let, 𝑎⃗ = (2, −3,4), 𝑏⃗⃗ = (−1,2,5), 𝑐⃗ = (3,6, −1); Find the following:
⃗⃗
𝑎⃗⃗.𝑏
i) 𝑎⃗. (𝑎⃗ + 𝑏⃗⃗ + 𝑐⃗) , ii) 𝑎⃗. (4𝑏⃗⃗) , iii) (𝑐⃗. 𝑏⃗⃗) 𝑎⃗ , iv) ( ⃗⃗ ⃗⃗) 𝑏⃗⃗
𝑏.𝑏

2) For which values of bi does the system have infinitely many solutions?
x1+3x2+5x3+8x4=b1
2x1+5x2+14x3+10x4=b2
x1+3x2+16x3-3x4=b3
2x1+6x2+20x3+6x4=b4

3) Find the inverse of the following matrices:


1 3 4 2 4 1
𝐴 = [3 11 18], 𝐵 = [6 6 −20]
2 6 11 1 5 12

4) Find det[A], det[B]:


4 1 3 0 3 2 11 7
8 1 12 5 1 −2 2 4
𝐴=[ ], 𝐵 = [ ]
−8 3 −8 28 2 0 7 8
4 4 6 26 1 −6 0 9

5) Evaluate, (|𝑢 ⃗⃗⃗ and (𝑣⃗x𝑤


⃗⃗|𝑣⃗). 𝑤 ⃗⃗⃗) + 2𝑢
⃗⃗ , where:
⃗⃗ = 𝑖⃗ − 2𝑗⃗ + 4𝑘⃗⃗
𝑢
𝑣⃗ = 2𝑖⃗ + 3𝑗⃗ − 𝑘⃗⃗
⃗⃗⃗ = 5𝑖⃗ + 4𝑗⃗ + 𝑘⃗⃗
𝑤

1 −1 1 −5 0 −2
2 −1 3 −6 1 9
6) Let, 𝐴 = [ ], 𝑟[𝐴] =?
1 0 2 −1 0 3
−1 2 0 9 −1 −1

7) For which values of the parameters ‘a’ and ‘b’ the system does have either i-unique solution
or ii-no solution, where:
x+y+z=3
x+y+(1+2b+ab)z=3+2b
(2+a)y+abz+b=b

8) Find the angle between the given vectors:


𝑎⃗ = (2,4,0), 𝑏⃗⃗ = (−1, −1,4)

9) Let, 𝑎⃗x𝑏⃗⃗ = 4𝑖⃗ − 3𝑗⃗ + 6𝑘⃗⃗ and 𝑐⃗ = 2𝑖⃗ + 4𝑗⃗ − 𝑘⃗⃗ , Find the corresponding scalar or vector:
(𝑎⃗x𝑏⃗⃗). 𝑐⃗=?

10) Prove the following statements true or false:


i)𝑎⃗x(𝑏⃗⃗x𝑐⃗) = (𝑎⃗x𝑏⃗⃗)x𝑐⃗ , ii)(𝑎⃗ + 𝑏⃗⃗)x(𝑎⃗ − 𝑏⃗⃗) = 2𝑏⃗⃗x𝑎⃗

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MIDDLE EAST TECHNICAL UNIVERSITY

11) Use Gauss Elimination Method to solve the following system:


2x1+2x2=0
-2x1+x2+x3=0
3x1+x3=0

12) Use Gauss-Jordan Elimination Method to solve the following system:


x1-2x2+x3=2
3x1-x2+2x3=5
2x1+x2+x3=1

13) Check whether the following set of vectors are linearly dependent:
i) 𝑢
⃗⃗1 = (1,2,3), 𝑢
⃗⃗2 = (1,0,1) , 𝑢
⃗⃗3 = (1, −1,5)
ii) 𝑢
⃗⃗1 = (2,6,3), 𝑢
⃗⃗2 = (1, −1,4) , 𝑢⃗⃗3 = (3,2,1) , 𝑢
⃗⃗4 = (2,5,4)

2 3 4
14) Let, 𝐴 = [ 1 −1 2] , Find: 𝑀12 , 𝑀32 , 𝐶13 , 𝐶22
−2 3 5

15) Find the determinants of the following matrices by cofactor expansion for the given systems:
0 2 0 5 0 0
𝐴 = [3 0 1] , 𝐵 = [0 −3 0]
0 5 8 0 0 2

16) Find the determinant of the following system by using upper or lower triangular matrix form:
1 1 5
𝐴 = [ 4 3 6]
0 −1 1

17) Find the inverse of the following matrix:


0 2 0
𝐴 = [ 0 0 1]
8 0 0

18) Find the inverse of the following matrix by using Gauss-Jordan Method:
1 2 3
𝐴 = [ 4 5 6]
7 8 9

19) Find 𝑥⃗ for the following system if; 𝐴𝑥⃗ = 𝐵 and 𝐴−1 𝐴𝑥⃗ = 𝐴−1 𝐵 :
x1+x2=4
2x1-x2=14

20) Solve the following system by using Cramer’s Rule:


-3x1+x2=3
2x1-4x2=-6

Directions:
-Solve each problem clearly with explicitly showing each step.
-The due date will not be extended. The portal will not accept late submissions.
-Problems are solely for you to solve.
-Solutions should be submitted as a single PDF document, each problem should clearly be numbered
and can easily be read and evaluated. Please check the neatness.
-Name each document according to: <Name_LastName_ID>

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