# Thread: Prove that a matrix is non-singular

1. ## Prove that a matrix is non-singular

Hey, i am really stuck with this question and any help would be greatly appreciated!

If D is a diagonal matrix with positive diagonal entries and L is a lower triangular matrix, I have to show that D-L is non-singular.

2. Originally Posted by bobby
Hey, i am really stuck with this question and any help would be greatly appreciated!

If D is a diagonal matrix with positive diagonal entries and L is a lower triangular matrix, I have to show that D-L is non-singular.

This is not true unless $\displaystyle L$ is a strictly lower triangular matrix. Otherwise take $\displaystyle D=L$ then then $\displaystyle D-L$ is a zero matrix which is definitely not invertible. However, if $\displaystyle L$ is a stricly lower triangular matrix then $\displaystyle D-L$ is a lower triangular matrix with non-zero diagnol entries. Now use the fact that the determinant of a triangular matrix is the product of its entires diagnol entries. Therefore, the determinant is non-zero.