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1. (10 points) Find the complete solution of Ax=b:

[1 1

-1

1

X=

bi

b2

2. (4 points) What is the rank of the system matrix D?

D

1 -3

-3 9

2 -6

12

4

5

-3

=

2

-1

4

2

3. 1 = 3 is an eigenvalue of the system matrix:

4 2

-1 1

24

-3

9

(a) (10 points) Calculate all eigenvalues of the system.

(b) (6 points) Calculate the eigenvector(s) for the eigenvalue 1 = 3.

If the sixth number of your student number ***X) is 5-9 then:

V3 1

F=

2 2

1 V3

2

(a) (4 points) Calculate the eigenvalues of the system matrix F.

(b) (5 points) The transformation x + Ax is the composition of a rotation and a

scaling.

• Find the rotation o, where a 5057.

• Find the scaling factor r.

4. In this question 2 variables A and I need to be used, they are determined by the fifth

number of your student number (****X*):

Table 1: Variables exercise 4

fifth digit|0|1| 2 | 3 | 4 | 5 | 6 | 7 8 9

A 0 0 -1 -1 -1 0 0 0 1 1

S2 55 5544433

5

2.11 -02 +2.63 = 1

-6×1 + 1.22 +2.13 = 0

8.01 22 +2.13 = 4

(a) (15 points) Find the inverse of A.

(b) (8 points) Solve x for Ax = b.

5. Given the three points A(2,3), B(-1,2) and C(-2,4).

(a) (4 points) Write down the vector equation of the line L through B and C.

(b) (8 points) Determine the normal vector that is perpendicular to line L, and points

towards point A.

(c) (7 points) Calculate the reflection of A over line L while using a transformation

matrix.

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Explanation & Answer:

5 Questions

Tags:

equation of the line

System matrix

eigenvalue of the system matrix

eigenvector of the eigenvalue

rotation and scaling matrix

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