Consider a matrix $A$ of dimension $4\times 4$ $$ A=\begin{pmatrix} a_{1} & a_2 & a_3 & a_4\\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ \end{pmatrix} $$ such that $a_1+a_2+a_3+a_4=1$ and $0\leq a_i\leq 1$ $\forall i \in \{1,2,3,4\}$.
Could you help me to show that this matrix is not invertible? In case the proof is too long, an intuition of the argument would be sufficient.
It's an upper triangular matrix, so you can trivially calculate the determinant by multiplying the diagonals, which gives you a determinant of 0 meaning the matrix is non-invertible.