Matrices and Transforming Geometric Shapes (Grades 11-12)

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Matrices and Transforming Geometric Shapes

1. 
Given the triangle with vertices [math](1,1), (2,3),[/math] and [math](5,1)[/math], which of the following matrix expressions would represent the reflection of this triangle over the y-axis?
  1. [math] [[1,0],[0,-1]] \ [[1,1],[2,3],[5,1]] [/math]
  2. [math][[-1,0],[0,1]] \ [[1,2,5],[1,3,1]][/math]
  3. [math] [[-1,0],[0,1]] \ [[1,1],[2,3],[5,1]] [/math]
  4. [math][[1,0],[0,-1]] \ [[1,2,5],[1,3,1]][/math]
2. 
The vertex matrix for rectangle ABCD is [math]V = [[1,1,4,4],[1,3,3,1]] [/math]. Which of the following is the correct transformed vertex matrix, if the transformation matrix [math]A = [[0,1],[-1,0]] [/math] is applied?
  1. [math] [[1,-1],[3,-1],[3,-4],[1,-4]] [/math]
  2. [math] [[-1,-3,-3,-1],[1,1,4,4]] [/math]
  3. [math] [[1,3,3,1],[-1,-1,-4,-4]] [/math]
  4. [math] [[-1,-3,-3,-1],[-1,-1,-4,-4]] [/math]
3. 
For the vertex matrix [math]V = [[1,1,3,3],[-1,2,2,-1]][/math] and transformation matrix [math]M = [[1,2],[0,1]][/math], what is the resulting shape if M is applied to V? Choose the most specific classification possible.
  1. Square
  2. Parallelogram
  3. Rectangle
  4. Quadrilateral
4. 
Put the vertices of the shape in the graph below into a vertex matrix, and then apply the rotation matrix [math]A = [[sqrt(2)/2, -sqrt(2)/2],[sqrt(2)/2, sqrt(2)/2]][/math]. By what angle, measured in degrees, has the shape been rotated (given that the angle of rotation is counterclockwise)?
T Q1 Horizontal Inverted
  1. 30°
  2. 45°
  3. 60°
  4. 105°
5. 
Which of the following equations correctly state that, for a figure in the cartesian plane, a reflection over the x-axis, followed by a reflection over the y-axis, is identical to a 180° rotation (either clockwise or counterclockwise)?
  1. [math][[1,0],[0,1]] [[-1,0],[0,-1]] = [[-1,0],[0,-1]][/math]
  2. [math][[1,0],[0,-1]] [[-1,0],[0,1]] = [[0,1],[1,0]][/math]
  3. [math] [[-1,0],[0,1]] [[1,0],[0,-1]] = [[-1,0],[0,-1]][/math]
  4. [math][[1,0],[0,-1]] [[-1,0],[0,1]] = [[-1,0],[0,-1]][/math]
6. 
Which one of the following matrix equations correctly shows the transformation of the first rectangle, whose vertex matrix is [math]V_1 = [[1,1,4,4],[1,2,2,1]][/math], to the second rectangle, whose vertex matrix is [math]V_2 = [[-4,-4,2,2],[1,3,3,1]] ?[/math]
1x3 Q1 Horizontal
2x6 Q2 Horizontal
  1. [math]V_2 = [[2,0],[0,2]] (V_1 - [[1,1,1,1],[1,1,1,1]] ) - [[4,4,4,4],[-1,-1,-1,-1]] [/math]
  2. [math]V_2 = [[2,0],[0,2]] V_1 - [[5,5,5,5],[0,0,0,0]] [/math]
  3. [math]V_2 = [[-4,1/2],[3/2, 1]] V_1 [/math]
  4. [math]V_2 = [[2,2],[2,2]] ( V_1 - [[1,1,1,1],[1,1,1,1]]) [/math]
7. 
For the following questions, let [math]T = [[2,-2],[0,3]] [/math] be a transformation matrix. Also, let [math] V_1 = [[2,5,5,2],[2,2,-1,-1]] [/math] be the vertex matrix of a square whose area is 9 units squared.
A. 
What is the resulting matrix if [math]V_1[/math] is transformed by [math]T[/math]? Let this matrix be [math]V_2[/math].
  1. [math][[0,6,12,6],[6,6,-3,-3]] [/math]
  2. [math][[0,6,5,2],[6,6,-1,-1]][/math]
  3. [math][[2,6,8,2],[0,6,3,-1]][/math]
  4. [math][[4,10,10,4],[6,6,-3,-3]][/math]
B. 
What is the area of the resulting shape, defined by the vertex matrix [math]V_2 ?[/math]
  1. 108 units squared
  2. 72 units squared
  3. 54 units squared
  4. 36 units squared
C. 
What is the absolute value of the determinant of the transformation matrix, [math]T ?[/math]
  1. 6
  2. 7
  3. 12
  4. 3
D. 
How does the absolute value of the determinant of the transformation matrix relate to the area of the two quadrilaterals (the square, [math]S_1[/math] and the transformed shape, [math]S_2[/math])? Choose the equation which correctly describes this relationship.
  1. [math]|det(T)| = "Area"(S_1) \ "Area"(S_2) [/math]
  2. [math]"Area"(S_2) = |det(T)| \ "Area"(S_1)[/math]
  3. [math]|det(T)| = |"Area"(S_2) - "Area"(S_1)| [/math]
  4. There is no relationship between the absolute value of the determinant of T and the area of the shapes.

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