$GL_n(\mathbb F_q)$ has an element of order $q^n-1$ > For fixed prime power $q$ show that the general linear group $GL_n(\mathbb F_q)$ of invertible matrices with entries in the finite field $\mathbb F_q$ has an element of order $q^n-1$. I tried to show this question with showing diagonal matrix but i can't find element directly competible with order i think i am on wrong way please give me clue ?

Hint: Realize $\mathbb{F}_{q^n}^*$ as a subgroup of $\mathrm{GL}_n(\mathbb{F}_q)$.