Matrices：練習問題

Find the determinant of the matrix $M = \begin{pmatrix} 2 & 1 & 3 \\ 4 & -1 & 2 \\ 1 & 0 & 5 \end{pmatrix}$.

The matrix $$M = \begin{pmatrix} 3 & 2 \\ 1 & 4 \end{pmatrix}$$ transforms points in the $x-y$ plane. Which of the following is an invariant line through the origin under this transformation?

Given the matrix \(A = \begin{pmatrix} 2 & 1 \\ 0 & 3 \end{pmatrix}\). Use the Cayley-Hamilton theorem to find an expression for \(A^3\) in the form \(pA + qI\), where \(I\) is the \(2 \times 2\) identity matrix.

Given matrices $$A = \begin{pmatrix} 2 & 1 \\ 3 & 4 \end{pmatrix}$$ and $$B = \begin{pmatrix} 0 & 3 \\ 1 & 2 \end{pmatrix}$$, find the matrix product $$AB$$.

Consider the system of linear equations:
$$x + 2y - z = 1$$
$$2x + 3y + z = 2$$
$$x + y + 2z = k$$
Which of the following statements about the solution to this system is correct?

Given the matrix $A = \begin{pmatrix} 3 & 1 \\ 5 & 2 \end{pmatrix}$, find its determinant.

The matrix $M$ is given by $M = \begin{pmatrix} 4 & 5 \\ 3 & 4 \end{pmatrix}$. Find the inverse matrix $M^{-1}$.

Find the eigenvalues of the matrix \( A = \begin{pmatrix} 4 & 1 \\ 2 & 3 \end{pmatrix} \).

Given the matrices $A = \begin{pmatrix} 2 & 1 \\ 3 & 4 \end{pmatrix}$ and $B = \begin{pmatrix} 1 & -2 \\ 0 & 5 \end{pmatrix}$.

(a) Find the matrix $2A - B$.

(b) Calculate the determinant of matrix $A$.

A linear transformation $T$ in the x-y plane is represented by the matrix $M = \begin{pmatrix} 5 & -1 \\ 2 & 2 \end{pmatrix}$.

(a) Find the characteristic equation of $M$.

(b) Find the eigenvalues of $M$.

(c) For each eigenvalue, find a corresponding eigenvector.