3. Let $A$ be a 3x3 square matrix, with all elements from the set $\{-1,1\}$.

a) Determine the possible values of the determinant of matrix $A$.

b) Prove that the matrix $A^{n}, n \in \mathbb{N}^{*}$, has all non-zero elements.
3. a) If we add the first row to rows 2 and 3, on these rows there will be only even numbers. We factor out 2 from each of them and obtain that the determinant of matrix $A$, which is an integer, is divisible by 4.

However, $\operatorname{det} A$ is the sum of six terms, each equal to 1 or -1, so it has a value between -6 and 6. It follows that det $A \in \{-4,0,4\}$ and it is immediately apparent that all three values are possible.

b) It is easily demonstrated, by mathematical induction, that the matrix $A^{n}$ has all elements odd, for any $n \in \mathbb{N}^{*}$. In particular, all elements of $A^{n}$ will be non-zero.