Matrices and Transformation：練習問題

A linear transformation in 2-D is a reflection in the line \(y = -x\). Which of the following matrices represents this transformation?

A 2-D linear transformation consists of a rotation of \(90^\circ\) anticlockwise about the origin, followed by a stretch with scale factor 2 parallel to the \(x\)-axis. Find the single matrix that represents this combined transformation.

A linear transformation is represented by the matrix \(\mathbf{M} = \begin{pmatrix} 3 & 0 \\ 2 & 1 \end{pmatrix}\). Determine the full set of invariant lines for this transformation.

A 3-D shape with a volume of 5 cubic units is transformed by the matrix \(\mathbf{A} = \begin{pmatrix} 1 & 2 & 1 \\ 0 & 3 & 2 \\ 4 & 0 & 1 \end{pmatrix}\). Calculate the volume of the resulting image shape.

The matrix \(\mathbf{M} = \begin{pmatrix} a & 1 & 1 \\ 1 & b & 1 \\ 1 & 1 & 1 \end{pmatrix}\) is singular. Given that the point \(P(1, 2, 1)\) is mapped to the point \(P'(4, 6, 4)\) under the transformation represented by \(\mathbf{M}\), find the value of the constant \(a + b\).

Find the \( 2 \times 2 \) matrix representing a reflection in the line \( y = -x \).

A transformation is represented by the matrix \(\mathbf{M} = \begin{pmatrix} -1 & 4 \\ 2 & -3 \end{pmatrix}\). Find the area of the image of a shape with an initial area of 10 square units and state whether the transformation reverses its orientation.

A 3-D linear transformation is defined by the matrix \(\mathbf{A} = \begin{pmatrix} 1 & -2 & 2 \\ 0 & 4 & k \\ 2 & 1 & 1 \end{pmatrix}\). Determine the value of the constant \(k\) for which the transformation is singular, and find the two values of \(k\) for which the volume scale factor is \(15\).

A linear transformation R is a rotation through \(135^\circ\) counter-clockwise about the origin. A second linear transformation S is a stretch parallel to the \(x\)-axis with scale factor \(k\), where \(k > 0\).
(a) Write down the matrices \(\mathbf{M}_R\) and \(\mathbf{M}_S\) representing the transformations R and S respectively.
(b) The combined transformation T consists of R followed by S. Find the matrix \(\mathbf{M}_T\) representing T in terms of \(k\).
(c) A triangle with area \(12\) square units is transformed by T. Given that the area of the image triangle is \(12\sqrt{2}\) square units, determine the value of \(k\).

A linear transformation in 3-D space is represented by the matrix \(\mathbf{M} = \begin{pmatrix} 1 & a & 1 \\ 0 & 2 & 1 \\ 1 & 1 & 1 \end{pmatrix}\), where \(a\) is a constant.
(a) Find \(\det(\mathbf{M})\) in terms of \(a\).
(b) State the value of \(a\) for which the matrix \(\mathbf{M}\) is singular.
(c) Given that \(a = 2\), find the inverse matrix \(\mathbf{M}^{-1}\) without using a calculator.
(d) For the case \(a = 2\), find the coordinates of the invariant points of the transformation.