How to find all values of for a given homogeneous system that has a nontrivial solution from a given matrix?

Question

The full question: How to find all values for which the homogeneous system (₂ − ) x= 0 has a nontrivial solution?

The given matrix:

We know that any solution in which at least one variable has a nonzero value is called a nontrivial solution.

Now, my problem is that how can I utilize with I₂ on the problem? Especially since these two variables made me so confused on how I can solve the values for , using also the given matrix A.

Your responses would indeed help me a lot since I am very new with trivial and nontrivial solutions that involve matrices. Thank you very much!

Answer 1

For the system $(\lambda I_2-A)x=0$ to have a non-trivial solution i.e. $x\ne 0$, you simply need
$det(\lambda I_2-A)=0$
$\implies det\begin{pmatrix}\lambda+1& -2\\-2&\lambda-2\end{pmatrix}=0$
$\implies \lambda^2-\lambda-6=0$
$\implies \lambda=3,-2$

Answer 2

We want to find all values $\lambda$ for which the homogeneous system $(\lambda I_2 − ) = 0$ has a nontrivial solution.

The matrix is given as

$$A = \begin{bmatrix} -1 & -2 \\ -2 & 2 \end{bmatrix}$$

Many problems present themselves in terms of an eigenvalue problem:

$$A·v=\lambda·v \\\lambda.v -A.v= 0 \\ \lambda·I_2·v - A·v=0 \\ (\lambda·I_2 - A)·v=0$$

If $v$ is non-zero, this equation will only have a solution if

$$|\lambda.I_2 - A|=0$$

This equation is called the characteristic equation of $A$ and is an $n^{th}$ order polynomial in $\lambda$ with $n$ roots, where these roots are called the eigenvalues of $A$.

We have

$$|\lambda.I_2 - A|= \begin{vmatrix} \lambda + 1 & 2 \\ 2 & \lambda -2 \end{vmatrix} = 0$$

Can you proceed?