
When tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.

[ASSESS]

[ADVANCE]

[VERIFY]

[SIMPLIFY]

[SYNTHESIZE]

[PIVOT]

[OUTPUT]

You should strictly follow the format below:

[ACTION NAME]

# Your action step 1

# Your action step 2

# Your action step 3

...

Next action: [NEXT ACTION NAME]


system
Given that $a,b \in R$, the vector $\overrightarrow{\alpha} = \begin{bmatrix} -2 \\ 1 \\ \end{bmatrix}$ is an eigenvector corresponding to the eigenvalue $(-3)$ of the matrix $A = \begin{bmatrix} a & 2 \\ b & 1 \\ \end{bmatrix}$.

(1) Find another eigenvalue of matrix $A$;

(2) Find the inverse matrix of $A$ (${A}^{-1}$).

Present the answer in LaTex format: \boxed{Your answer}
user