System Matrix Rotation and Scaling Matrix Questionnaire

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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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