Find all values of 𝜆 for a given homogeneous system that has a nontrivial solution from a given matrix

Question

The full question: How to find all values 𝜆 for which the homogeneous system (𝜆𝐼₂ − 𝐴) 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 𝜆 with I₂ on the problem? Especially since these two variables made me so confused on how I can solve the values for 𝜆, 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
$det(\lambda I_2-A)=0$
$\implies det\begin{pmatrix}\lambda+1& -2\\-2&\lambda-2\end{pmatrix}=0$
$\implies \lambda^2-\lambda-6=0$
$\implies \lambda=3,-2$