Matrices and transformations: แบบฝึกหัด

Find the invariant points of the transformation represented by the matrix \( \mathbf{A} = \begin{pmatrix} 3 & -2 \\ 1 & 0 \end{pmatrix} \).

A transformation is represented by the matrix \( \mathbf{M} = \begin{pmatrix} 2 & 0 \\ 0 & 3 \end{pmatrix} \). A triangle with an area of 5 square units is transformed by \( \mathbf{M} \). What is the area of the image triangle?

Determine the matrix representing a shear parallel to the \( x \)-axis, mapping the point \( (0, 1) \) to \( (3, 1) \).

The transformation \(T\) is a shear parallel to the \(x\)-axis which maps the point \((1, 2)\) to the point \((5, 2)\). Find the \(2 \times 2\) matrix representing \(T\).

The matrix \(\mathbf{M} = \begin{pmatrix} 2 & k \\ 4 & 6 \end{pmatrix}\) represents a transformation that maps an area of 5 units onto an area of 10 units. Find the possible values of the constant \(k\).

A linear transformation is represented by the matrix \( \mathbf{M} = \begin{pmatrix} k & 2 \\ 3 & k-1 \end{pmatrix} \), where \( k \) is a constant.

(a) Find the values of \( k \) for which the matrix \( \mathbf{M} \) is singular.

(b) In the case where \( k = 4 \), find the coordinates of the invariant point, other than the origin, under the transformation represented by \( \mathbf{M} \).

(c) When \( k = 2 \), the transformation represented by \( \mathbf{M} \) maps a shape with area 5 square units onto an image shape. Find the area of the image shape.

The matrix \( \mathbf{M} \) represents a reflection in the line \( y = x \). The matrix \( \mathbf{N} \) represents a shear parallel to the \( x \)-axis such that the point \( (1, 1) \) is mapped to the point \( (3, 1) \).

(a) Find the matrices \( \mathbf{M} \) and \( \mathbf{N} \).

(b) Find the matrix \( \mathbf{P} = \mathbf{NM} \) and describe the single transformation represented by \( \mathbf{P} \).

(c) Find the matrix \( \mathbf{P}^{-1} \) and determine the area scale factor of the transformation represented by \( \mathbf{P} \).

(d) Find the equation of the invariant line (other than the line of invariant points) for the transformation represented by \( \mathbf{N} \).