Matrices：練習問題

Given the matrices \(\mathbf{A} = \begin{pmatrix} 2 & -1 \\ 3 & 4 \end{pmatrix}\) and \(\mathbf{B} = \begin{pmatrix} 5 & 0 \\ -2 & 1 \end{pmatrix}\), find the matrix \(\mathbf{A} + 3\mathbf{B}\).

A 2-D linear transformation is represented by the matrix \( \mathbf{M} = \begin{pmatrix} 3 & k \\ -1 & 2 \end{pmatrix} \). This transformation maps a unit square onto an image with area 10 and reverses the orientation of the shape. Find the value of the constant \( k \).

Find the equations of the two invariant lines passing through the origin for the linear transformation represented by the matrix \( \mathbf{M} = \begin{pmatrix} 2 & 1 \\ 1 & 2 \end{pmatrix} \).

The matrix \(\mathbf{A}\) is defined as \(\mathbf{A} = \begin{pmatrix} 1 & a & 0 \\ 0 & 2 & 1 \\ a & 0 & 1 \end{pmatrix}\). Given that the volume scale factor of the transformation represented by \(\mathbf{A}\) is 6 and the orientation is reversed, find the possible values of the constant \(a\).

The matrix \( \mathbf{A} \) is defined as \( \mathbf{A} = \begin{pmatrix} 2 & 1 \\ 5 & 3 \end{pmatrix} \). Which of the following is the inverse matrix \( \mathbf{A}^{-1} \)?

Find the positive value of the constant \( k \) for which the matrix \( \mathbf{M} = \begin{pmatrix} k & 4 \\ 3 & k-1 \end{pmatrix} \) is singular.
(1-2 sentence answer only)

Find the \( 2 \times 2 \) matrix that represents a stretch parallel to the \( y \)-axis with scale factor 3, followed by a reflection in the line \( y = x \).
(1-2 sentence answer only)

Determine the value of the constant \( k \) for which the following system of equations is consistent and has an infinite number of solutions:
\( x - y + 2z = 3 \)
\( 2x + y - z = 5 \)
\( 5x - 2y + 5z = k \).
(1-2 sentence answer only)

A linear transformation \( T \) in 2-D space is defined by the matrix \(\mathbf{M} = \begin{pmatrix} k & 2 \\ 3 & k-1 \end{pmatrix}\), where \( k \) is a real constant.
(a) Find the values of \( k \) for which the transformation \( T \) is singular. Geometrically describe what happens to the image of the unit square in these cases. [2]
(b) Given that \( k = 5 \), the transformation \( T \) maps a triangle with area 4 square units onto an image triangle. Calculate the area of the image triangle. [2]
(c) For \( k = 2 \), find the inverse matrix \(\mathbf{M}^{-1}\) and use it to find the coordinates of the point that is mapped by \( T \) to the point \( (4, 1) \). [2]

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

(a) Show that the determinant of \(\mathbf{M}\) is \(7 - k\). [2 points]

(b) Find the value of \(k\) for which the matrix \(\mathbf{M}\) is singular. Geometrically interpret the effect of the transformation on the volume of a solid when \(k\) takes this value. [1 point]

(c) In the case where \(k = 3\), find the inverse matrix \(\mathbf{M}^{-1}\). [3 points]

(d) Three planes are defined by the equations:
\(x + 2y - z = 4\)
\(2x + 3y + z = 13\)
\(x + y - 2z = -1\)
Use your answer to part (c) to solve this system of simultaneous equations. [1 point]

(e) When \(k = 7\), determine whether the system of equations represented by \(\mathbf{M}\mathbf{x} = \begin{pmatrix} 4 \\ 13 \\ -1 \end{pmatrix}\) is consistent. Hence, describe the geometrical arrangement of the three planes in this case. [1 point]