## Problem 3.

Let $G$ be the set of 3x3 square matrices whose elements are equal to -1 or 1.

a) How many elements does the set $G$ have?

b) Determine the set $\{\operatorname{det}(A) \mid A \in G\}$.
## Problem 3.

Let G be the set of 3x3 square matrices whose elements are equal to -1 or 1.

a) How many elements does the set G have?

b) Determine the set $\{\operatorname{det} A \mid A \in G\}$.

## Solution

a) A matrix in G has 9 elements, each of which can take 2 values (-1 or 1). The set G has $2^{9}=512$ elements.

b) Let $A \in G$. From the definition of a determinant of order 3, we obtain that det(A) is a sum of 6 terms of the form $( \pm 1) \cdot( \pm 1) \cdot( \pm 1)$, so $-6 \leq \operatorname{det}(A) \leq 6$.

By adding the first row to the second and third rows, we get $\operatorname{det}(A)=\operatorname{det}(B)$ where $B$ is a matrix that has -1 or 1 on the first row, and 0, -2, or 2 on the second and third rows. We factor out the common factor 2 from the second and third rows, and then $\operatorname{det}(A)=4 \operatorname{det}(C)$, where $C$ is a matrix that has -1 or 1 on the first row, and 0, -1, or 1 on the second and third rows. Since $\operatorname{det}(C) \in \mathbb{Z}$, it follows that 4 divides $\operatorname{det}(A)$, hence $\operatorname{det}(A) \in\{-4,0,4\}$. 3p

$$
\begin{aligned}
& \text { Let } A_{1}=\left(\begin{array}{lll}
1 & 1 & 1 \\
1 & 1 & 1 \\
1 & 1 & 1
\end{array}\right) \text { and } A_{2}=\left(\begin{array}{ccc}
1 & 1 & 1 \\
1 & -1 & -1 \\
1 & 1 & -1
\end{array}\right) . \text { Clearly } A_{1}, A_{2},-A_{2} \in G \quad \text { and } \operatorname{det}\left(A_{1}\right)=0, \operatorname{det}\left(A_{2}\right)=4 \text {, } \\
& \operatorname{det}\left(-A_{2}\right)=-4 \text {. Therefore, }\{\operatorname{det}(\mathbf{A}) \mid \mathbf{A} \in G\}=\{-4, \mathbf{0}, \mathbf{4}\} .
\end{aligned}
$$