# Lower and Upper Triangular Matrices

$A$ is an $n\times n$ matrix and $L$ is an $n \times n$ nonsingular lower triangular matrix. How can I prove that if $LA$ is lower triangular, then $A$ is lower triangular?

How can I do the same for upper triangular matrix, $B$ is $n\times n$ and $Z$ is $n\times n$ nonsingular upper triangular matrix. If $ZB$ is upper triangular, then $B$ is upper triangular?

• Do you know already that the inverse of a nonsingular lower triangular matrix is also lower triangular? If so, you can simply write $A=L^{-1}\cdot LA$. – Greg Martin Jan 25 '15 at 19:56
Given these facts, we can quickly deduce that if both $$L$$ and $$LA$$ are lower triangular, with $$L$$ nonsingular, then $$A = L^{-1}(LA)$$ is a product of two lower triangular matrices and thus is itself lower triangular.