線性方程組：练习题

Which of the following systems of linear equations has a unique solution?

If the system of linear equations \(kx + 3y = 1\) and \(3x + ky = 1\) has no solution, what is the value of \(k\)?

Consider the following system of linear equations:
\(x + y + z = 1\)
\(2x + y - z = 2\)
\(x + 2y + 4z = p\)
Find the value of \(p\) such that the system is consistent.

Consider a system of homogeneous linear equations:
\(ax + by = 0\)
\(cx + dy = 0\)
Which of the following conditions ensures that the system has only the trivial solution \((x, y) = (0, 0)\)?

For what value of \(m\) does the following system of homogeneous linear equations have non-trivial solutions?
\(x + y + z = 0\)
\(2x - y + 3z = 0\)
\(5x + 2y + mz = 0\)

If the augmented matrix of a system of linear equations in variables $x_1, x_2, x_3$ is row equivalent to the following matrix in row echelon form:

$$\begin{pmatrix} 1 & 2 & 1 & | & 5 \ 0 & 1 & -1 & | & 1 \ 0 & 0 & 1 & | & 2 \ \end{pmatrix}$$

What is the value of the variable $x_2$?

A system of linear equations is represented by the matrix equation $A\mathbf{x} = \mathbf{b}$, where $A = \begin{pmatrix} 3 & -1 \\ 1 & 2 \end{pmatrix}$ and $\mathbf{b} = \begin{pmatrix} 7 \\ 0 \end{pmatrix}$. Find the value of $x$ (the first component of $\mathbf{x}$).

Find the general solution to the following system of homogeneous linear equations, expressing the solution in terms of a parameter $t$:

$$\begin{aligned} x + y + z &= 0 \\ 2x + y + z &= 0 \\ 3x + 2y + 2z &= 0 \end{aligned}$$

Consider the following system of linear equations in \(x, y, z\):
\( \begin{cases} x + 2y + z = 4 \\ 2x + y + 3z = 7 \\ 3x + 3y + az = 11 \end{cases} \)
where \(a\) is a real constant.

(a) Find the value of \(a\) such that the system does not have a unique solution.

(b) Suppose \(a = 5\). Solve the system of equations for \(x, y, z\) using Gaussian elimination.

Consider the system of linear equations:

(a) Find the value(s) of $k$ for which the system does not have a unique solution.

(b) For the value(s) of $k$ found in (a), determine the nature of the solution (no solution or infinitely many solutions) for each case. Hence, find the general solution of the system for the value of $k$ that yields infinitely many solutions.