Properties of set $\{X∈ M_n(\mathbb{C}) | \operatorname{adj}(X)=I_n\}$

Question

"For $n≥4$, let $U=\{X∈ M_n(\mathbb{C}) | \operatorname{adj}X=I\}$. Which of the following statements are true/false? Justify:

1. $U$ contains $n-1$ elements,
2. $U$ contains only scalar matrices,
3. if $X_1,\, X_2∈U$, then $X_1X_2∈U$
4. if $X_1, X_2∈ U$, then $X_1+X_2∈ U$"

I honestly don't know how to begin. I suppose for the last 2 statements if you know that $U$ is a subspace, you could solve it straightforwardly, but I don't even know how to get to that point.

Do I need to use the fact that |A| A^-1 = adjA?

Answer 1

Clearly, such a amtrix must be invertible, so that $\det X \neq 0$. Start from the property of $\mathrm{adj} X $ $$X \cdot \mathrm{adj} X = (\det X) \cdot I$$ Since $\mathrm{adj} X=1$, you have $$X= (\det X) I$$ thus, $X$ is a scalar matrix. Taking determinant, you have $$\det X = (\det X)^n$$ thus $\det X = e^{\frac{2 k \pi i}{n-1}}$ must be equal to a root of unity for some $k \in \{ 1, \dots , n-1\}$.

After this, check that this condition is also sufficient. Thus 1,2,3 are true; 4 is false.