矩陣：練習題

A linear transformation \(T\) in the \(x-y\) plane is represented by the matrix \(\mathbf{M} = \begin{pmatrix} a & 2 \\ 3 & b \end{pmatrix}\). Given that the point \((1, 2)\) is an invariant point under \(T\), find the values of \(a\) and \(b\).

The matrix \(\mathbf{B} = \begin{pmatrix} 2 & 1 \\ 0 & 3 \end{pmatrix}\). Find the matrix \(\mathbf{M}\) such that \(\mathbf{M} = \sum_{k=1}^{n} \mathbf{B}^k\). What is the top-right entry of \(\mathbf{M}\)?

A linear transformation \(T\) in the \(xy\)-plane is represented by the matrix \(\mathbf{M} = \begin{pmatrix} 4 & -1 \\ 2 & 1 \end{pmatrix}\). Find the equations of the two invariant lines through the origin under the transformation \(T\).

Let \(\mathbf{A} = \begin{pmatrix} 1 & 2 & 1 \\ 2 & 1 & 0 \\ -1 & 0 & 1 \end{pmatrix}\). Find the determinant of \(\mathbf{A}^{-1} (2\mathbf{I})\), where \(\mathbf{I}\) is the \(3 \times 3\) identity matrix.

Given the matrix \(\mathbf{A} = \begin{pmatrix} 1 & -1 & 1 \\ 0 & 2 & -1 \\ 2 & 3 & 0 \end{pmatrix}\), find the value of \(k\) such that \(\det(k\mathbf{A}) = -80\), given that \(\det(\mathbf{A}) = 5\).

Find the inverse of the matrix \(\mathbf{A} = \begin{pmatrix} 2 & 1 & 0 \\ 1 & -1 & 1 \\ 0 & 2 & -1 \end{pmatrix}\).

The transformation \(T\) in the \(x-y\) plane is represented by the matrix \(\mathbf{A} = \begin{pmatrix} 5 & -2 \\ 4 & -1 \end{pmatrix}\). Find the equation of the invariant line through the origin, other than the line \(y = 2x\).

The transformation \(T_1\) in the \(x-y\) plane is a reflection in the line \(y = x\), and the transformation \(T_2\) is a rotation about the origin through an angle of \(\frac{\pi}{2}\) anticlockwise.

(a) Write down the \(2 \times 2\) matrices \(\mathbf{A}\) and \(\mathbf{B}\) that represent \(T_1\) and \(T_2\) respectively.

(b) Find the single matrix \(\mathbf{C}\) that represents the combined transformation of \(T_1\) followed by \(T_2\).

(c) Describe fully the single geometric transformation represented by the matrix \(\mathbf{C}\).

The matrix \( \mathbf{M} \) is given by \( \mathbf{M} = \begin{pmatrix} 5 & -2 \\ 4 & -1 \end{pmatrix} \).

(a) Find the invariant points of the transformation represented by \( \mathbf{M} \).
(b) Find the equations of the two invariant lines through the origin for this transformation.
(c) Show that the line with equation \( y = 2x + 3 \) is an invariant line under the transformation.