Homogeneous linear equations

Let $\mathbf{A}$ be an $n \times n$ matrix. Linear equations of the form $\mathbf{A} x = 0$ are called homogeneous.

If $\text{ det } \mathbf{A} \neq 0$ then $\mathbf{A} x = 0$ has exactly one solution, $x=0$. If $\text{ det } \mathbf{A} = 0$ then $\mathbf{A} x = 0$ has infinitely many solutions.

Practice problems

For what value of $c$ is there a nonzero solution to the following equation? For that value of $c$, find all solutions to the equation. $$\begin{pmatrix}1&1\\2&c\end{pmatrix}\begin{pmatrix}x\\y\end{pmatrix} = \begin{pmatrix}0\\0\end{pmatrix}$$
Solution
For what values of $\lambda$ are there nontrivial solutions to $$\begin{pmatrix}1&0&0\\0&2&0\\0&0&3\end{pmatrix}\begin{pmatrix}x\\y\\z\end{pmatrix} = \lambda \begin{pmatrix}x\\y\\z\end{pmatrix}$$
For each value, find a nontrivial solution.
Solution
Are there any real values of $c$ for which there is a nontrivial (nonzero) solution to $$\begin{pmatrix}1&c\\-c&2\end{pmatrix}\begin{pmatrix}x\\y\end{pmatrix} = \begin{pmatrix}0\\0\end{pmatrix}?$$
Solution

Homogeneous linear equations

Related topics

Practice problems