# Number of elements in the set of invertible lower triangular matrices over a finite field

Problem:

Let $$F_q$$ be a finite field with $$q$$ elements.

$$T_n(F_q) := \{ A = (a_{ij}) \in F^{n \times n}$$ | $$a_{ij} = 0$$ for $$i < j,$$ and $$a_{ij} \neq 0$$ $$\forall i \}$$.

Determine the number of elements in $$T_n(F_q)$$.

My solution is as follows:

Starting with the last row going upwards, there are:

$$q-1$$ possibilities for the last row;

$$(q-1)q$$ possibilities for the row before the last;

.

.

.

$$(q-1)q^{n-1}$$ possibilities for the first row.

Therefore, in total there are $$(q-1)^nq^{\sum_{i=1}^{n-1} i} = (q-1)^nq^{\frac{n(n-1)}{2}}$$ elements.

Could you, please, check my solution?