Why can't a triangular matrix with only zeros in its diagonal be invertible?
I know that it is not invertible but I don't know well the reasons, perhaps. Actually, I read that can't have any zero in its diagonal, in order to be invertible.
In general, I know that if a matrix has an inverse, the product of that matrix with its inverse is equal to the identity matrix, namely, with a matrix with 1s as its diagonal.
Best Answer
A upper triangular matrix with 0's on its diagonal has its first column filled with 0's. Therefore its determinant is 0, which means it is not invertible. For a lower triangular matrix, the same holds with the last column.
In case you're not familiar with determinants:When A is upper triangular, let's suppose $A^{-1}$ exists. Then $A^{-1}A=I$. Let $B=\begin{pmatrix} 1 \\ 0\\0\\...\\0 \end{pmatrix}$. Then $AB=0$. However, $(A^{-1}A)B=I B=B$ , but $$A^{-1}(AB)=A^{-1}0=0\ne B$$ We have a contradiction, therefore A is not invertible.