True or flase? If the product of two matrices is invertible then each matrices in the product is invertible.
It was said to be true but I didn't understood the argument: a matrix A \begin{pmatrix} 1 & 1 & 1\\ 1 & 0 & 1\\ 1 & 1 & 1 \end{pmatrix}
is non-invertible, its determinant equals $0$, furthermore it is equal to the product of diagonal coefficients of $A$, $A$ not being triangular.
Okay but $A*A$ isn't invertible neither, it gives:
\begin{pmatrix} 3 & 2 & 3\\ 2 & 2 & 2\\ 3 & 2 & 3 \end{pmatrix}
So I don't undertand the point, it should have demonstrate that $A*A$ was invertible and that $A$ wasn't.