# Find the complete solution of Ax

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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