# Math Help - 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 $L$ is a strictly lower triangular matrix. Otherwise take $D=L$ then then $D-L$ is a zero matrix which is definitely not invertible. However, if $L$ is a stricly lower triangular matrix then $D-L$ 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.

I missed out a part of the question which may be crucial as L is just lower triangular.

Suppose A is positive definite. Note that we can write A = D - L - L* with D and L defined in my first post and L* is the conjugate transpose of L i think. Show that D - L is non-singular.

Thanks very much!