Proposition

Let $V$ be a Finite Dimensional $\mathbb{F}$-vector space and $T\in\mathscr{L}(V)$. If there is a Finite Basis of $V$ such that $[a_{ij}]=\mathcal{M}(T)$ is upper-triangular with respect to this basis, then $T$ is invertible if and only if $a_{i,i}\not=0 \ \forall i=1,\cdots,n$.