$$\begin{matrix} * & * & *&*&* \\ *&*&*&*&*\\ 0&0&0&*&* \\ 0&0&0&*&* \\ 0&0&0&*&* \\ \end{matrix}$$ all the asterisk's belong to real numbers
The question is show $determinant=0$ by contemplating the Leibniz formula.
I know its zero intuitively and aware it has $5!$ terms in the formula but how can i prove it without calculating.
any help appreciated
Consider an arbitrary, fixed summand in Leibniz's formula. It must contain an element from each of the last three rows of the matrix. But these elements have to be in three distint columns. So one of them will be in one of the first three columns, thus zero.