Engineering analysis discussion questions

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When you give solutions, make sure you give step-by-step procedures and explanations to show your reasoning and justifications.

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1. (Points 10) Matrix A is singular: True or False? justify.

2. (Points 20) Find A – B, B + A + +,

3. (Points 20) Is it true: AB = BA? justify.

4. (Points 30) -.X = d, find X using Gauss-Jordan elimination method.

• Step 1: Write down the system of linear equations

• Step 2: Construct the augmented coefficient matrix.

• Step 3: Apply Gauss-Jordan elimination.

• Step 4: Reduced to row-echelon form.

5. (Points 20) Find the inverse of matrix B using Gauss-Jordan elimination method.

HW 2 EGN 3420 Engineering Analysis Deadline: 02/22/2019 (Friday) 11:59 PM &' 1 0 2 2 0 1 1 & A = !1 2 0%, B = !2 1 0%, X = ! ( %, d = !−1% &) 0 1 2 0 2 1 0 1. (Points 10) Matrix A is singular: True or False? justify. 2. (Points 20) Find A – B, B + A + +, 3. (Points 20) Is it true: AB = BA? justify. 4. (Points 30) -. X = d, find X using Gauss-Jordan elimination method. • Step 1: Write down the system of linear equations • Step 2: Construct the augmented coefficient matrix. • Step 3: Apply Gauss-Jordan elimination. • Step 4: Reduced to row-echelon form. 5. (Points 20) Find the inverse of matrix B using Gauss-Jordan elimination method.

seniorlecturerken
School: Purdue University

Hey there. I've managed to solve the problems. Kindly go through them and let me know if any of the solutions isn't clear.

Concepts:
1.
2.
3.
4.
5.
6.

Identity matrices
Gauss Jordan elimination method
Matrix transposition
Inverse of a matrix
Matrix multiplication

Question 1
Solution
A matrix is singular if and only if the determinant is zero.
𝑎11 𝑎12
For a matrix A = [𝑎21 𝑎22
𝑎31 𝑎32

𝑎13
𝑎23]
𝑎33

The determinant can be found by rewriting the matrix in the below form where column 1 and 2
are repeated;
𝑎11 𝑎12
[𝑎21 𝑎22
𝑎31 𝑎32

𝑎13 𝑎11
𝑎23 𝑎21
𝑎33 𝑎31

𝑎12
𝑎22]
𝑎32

The determinant is then obtained by subtracting the lagging product sum from the leading
product sum as shown below;
Determinant = (a11*a22*a33 + a12*a23*a31 + a13*a21*a32) –
(a31*a22*a13 + a32*a23*a11 + a33*a21*a12)
1 0 2
For the matrix A = [1 2 0]
0 1 2
The matrix above can be rewritten as;
1 0 2 1 0
[1 2 0 1 2 ]
0 1 2 0 1
The determinant is therefore given by;
Det = (1*2*2 + 0*0*0 + 2*1*1) – (0*2*2 + 1*0*1 + 2*1*0)
Det = (4 + 0 + 2 ) – (0 + 0 + 0)
Det = 6

Since the determinant of matrix A is not zero. The matrix is therefore not singular i.e the
statement is false as justified above

Question Two
Finding A – B

I.

For two matrices A and B their difference can be expressed as below;
𝑎11 𝑎12
[𝑎21 𝑎22
𝑎31 𝑎32

𝑎13
𝑏11
𝑎23] - [𝑏21
𝑎33
𝑏31

𝑏12
𝑏22
𝑏32

1
For A = [1
0

0 2
2 0
2 0] and B = [2 1
1 2
0 2

1
A – B = [1
0

0 2
2
2 0] - [2
1 2
0

II.

𝑏13
𝑎11 − 𝑏11
𝑏23] = [𝑎21 − 𝑏21
𝑏33
𝑎31 − 𝑏31

𝑎12 − 𝑏12
𝑎22 − 𝑏22
𝑎32 − 𝑏32

𝑎13 − 𝑏13
𝑎23 − 𝑏23]
𝑎33 − 𝑏33

1
0]
1

0 1
−1 0 1
1 0] = [−1 1 1]
2 1
0 −1 1

Finding B + A + I3

Note that I3 is a 3x3 identity matrix which can be expressed as;
1
I3 = [0
0

0 0
1 0]
0 1

Therefore B + A + I3 will be given by;
2 0 1
1 0 2
1 0
B + A + I3 = [2 1 0] + [1 2 0] + [0 1
0 2 1
0 1 2
0 0
4 0 3
B + A + I3 = [3 4 0]
0 3 4

2+1+1 0+0+0
0
0] = [2 + 1 + 0 1 + 2 + 1
0+0+0 2+1+0
1

1+2+0
0 + 0 + 0]
1+2+1

Question Three
Consider two matrices;
𝑎11 𝑎12 𝑎13
𝑏11
A = [𝑎21 𝑎22 𝑎23] 𝑩 = [𝑏21
𝑎31 𝑎32 𝑎33
𝑏31

AB =
𝑎11 ∗ 𝑏11 + 𝑎12 ∗ 𝑏21 + 𝑎13 ∗ 𝑏31
[𝑎21 ∗ 𝑏11 + 𝑎22 ∗ 𝑏21 + 𝑎23 ∗ 𝑏31
𝑎31 ∗ 𝑏11 + 𝑎32 ∗ 𝑏21 + 𝑎33 ∗ 𝑏3...

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Anonymous
Excellent job

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