矩陣變換：练习题

Which of the following matrices represents a linear enlargement about the origin with a scale factor of 3?

The matrix \( T = \begin{pmatrix} 2 & k \\ -1 & 4 \end{pmatrix} \) transforms a triangle with area 5 units\(^2\) to a triangle with area 45 units\(^2\). Given that \( k \) is a positive constant, find the value of \( k \).

A linear transformation \(T_1\) is a rotation through \(90^\circ\) clockwise about the origin. A second linear transformation \(T_2\) is a reflection in the line \(y = -x\). Which of the following geometric transformations is equivalent to the combined transformation of \(T_1\) followed by \(T_2\)?

A linear transformation \( S \) consists of a reflection in the \( x \)-axis followed by a rotation of \( 90^\circ \) anticlockwise about the origin. Find the single matrix that represents \( S \).

A linear transformation \( M \) maps the unit square with vertices \( (0,0), (1,0), (0,1), \) and \( (1,1) \) to a shape with vertices \( (0,0), (2,3), (-1,4), \) and \( (1,7) \) respectively. Find the matrix \( M \).

State the geometric transformation represented by the matrix \( \mathbf{M} = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} \).

A triangle \( T \) has an area of \( 12 \text{ cm}^2 \). Find the area of the image of \( T \) after it has been transformed by the matrix \( \mathbf{A} = \begin{pmatrix} 3 & 1 \\ 2 & 4 \end{pmatrix} \).

The matrix \( \mathbf{P} \) represents a reflection in the line \( y = x \) and the matrix \( \mathbf{Q} \) represents a rotation of \( 90^\circ \) anticlockwise about the origin. Find the single matrix that represents the transformation \( \mathbf{P} \) followed by \( \mathbf{Q} \).

The linear transformation \(T\) is represented by the matrix \( {A} = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} \), and the linear transformation \(U\) is represented by the matrix \( {B} = \begin{pmatrix} 3 & 0 \\ 0 & 3 \end{pmatrix} \).
(a) Describe geometrically the transformations represented by \( {A} \) and \( {B} \).
(b) Find the matrix \( {C} \) that represents the transformation \(T\) followed by \(U\).
(c) A triangle with area 5 square units is transformed by \( {C} \). Find the area of the image triangle.