This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
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Here are the descriptions of the transformations for each matrix, using the unit square with vertices O(0,0), A(1,0), B(0,1), and C(1,1). a) For the matrix 0 & 1 \\ 1 & 0 : Step 1: Apply the transformation to the vertices of the unit square. O' = 0 & 1 \\ 1 & 0 0 \\ 0 = 0 \\ 0 A' = 0 & 1 \\ 1 & 0 1 \\ 0 = 0 \\ 1 B' = 0 & 1 \\ 1 & 0 0 \\ 1 = 1 \\ 0 C' = 0 & 1 \\ 1 & 0 1 \\ 1 = 1 \\ 1 Step 2: Describe the transformation. The original vertices are (0,0), (1,0), (0,1), (1,1). The transformed vertices are (0,0), (0,1), (1,0), (1,1). This transformation swaps the x and y coordinates. The transformation is a reflection about the line y=x. b) For the matrix 0 & -1 \\ 1 & 0 : Step 1: Apply the transformation to the vertices of the unit square. O' = 0 & -1 \\ 1 & 0 0 \\ 0 = 0 \\ 0 A' = 0 & -1 \\ 1 & 0 1 \\ 0 = 0 \\ 1 B' = 0 & -1 \\ 1 & 0 0 \\ 1 = -1 \\ 0 C' = 0 & -1 \\ 1 & 0 1 \\ 1 = -1 \\ 1 Step 2: Describe the transformation. The original vertices are (0,0), (1,0), (0,1), (1,1). The transformed vertices are (0,0), (0,1), (-1,0), (-1,1). This corresponds to rotating the x-axis to the y-axis and the y-axis to the negative x-axis. The transformation is a rotation of 90^ counter-clockwise about the origin (0,0). c) For the matrix 2 & 0 \\ 0 & 2 : Step 1: Apply the transformation to the vertices of the unit square. O' = 2 & 0 \\ 0 & 2 0 \\ 0 = 0 \\ 0 A' = 2 & 0 \\ 0 & 2 1 \\ 0 = 2 \\ 0 B' = 2 & 0 \\ 0 & 2 0 \\ 1 = 0 \\ 2 C' = 2 & 0 \\ 0 & 2 1 \\ 1 = 2 \\ 2 Step 2: Describe the transformation. The original vertices are (0,0), (1,0), (0,1), (1,1). The transformed vertices are (0,0), (2,0), (0,2), (2,2). All coordinates are multiplied by 2. The transformation is an enlargement (or dilation) with a scale factor of 2, centered at the origin (0,0). d) For the matrix 0 & -1 \\ -1 & 0 : Step 1: Apply the transformation to the vertices of the unit square. O' = 0 & -1 \\ -1 & 0 0 \\ 0 = 0 \\ 0 A' = 0 & -1 \\ -1 & 0 1 \\ 0 = 0 \\ -1 B' = 0 & -1 \\ -1 & 0 0 \\ 1 = -1 \\ 0 C' = 0 & -1 \\ -1 & 0 1 \\ 1 = -1 \\ -1 Step 2: Describe the transformation. The original vertices are (0,0), (1,0), (0,1), (1,1). The transformed vertices are (0,0), (0,-1), (-1,0), (-1,-1). This transformation maps (x,y) to (-y,-x). The transformation is a reflection about the line y=-x.