Proving that a triangular matrix is invertible

[Icheb]

I have a square triangular matrix with d_{ij} = 0 for all 1 \le j < i \le n. Now I have to prove that this matrix is only then invertible when d_{ii} \ne 0 for all 1 \le i \le n.

From what I know a matrix is only then invertible when its determinant does not equal 0. I also think that the determinant of a triangular matrix is dependent on the product of the elements of the main diagonal and if that's true, I'd have the proof. However this is also where I'm stuck since I don't know how to prove that. Could someone help me there?

[D H]

Hint: Formulate the determinant using determinant expansion by minors on the first column.

You should get a very compact expression that only involves the diagonal elements.

[radou]

I also think that the determinant of a triangular matrix is dependent on the product of the elements of the main diagonal and if that's true, I'd have the proof. However this is also where I'm stuck since I don't know how to prove that. Could someone help me there?

It's a simple proof. Just use the definition of the determinant.

[Icheb]

Ok, do I understand it correctly that if all elements of one line equal 0, the determinant equals 0? So I would just have to prove that the matrix is linear (Did I translate that word correctly?) and my proof would be complete? Or am I missing something?

[matt grime]

You're missing something. It is straightfoward to show from the definition of determinant by expansion that the determinant of a triangular matrix is the product of the diagonal elements. A row of zeroes is neither here nor there.

[Icheb]

But why wouldn't it suffice if I assume one element of the main diagonal to be 0, then I use Gaussian transformation to change the last line so it contains only 0s? Then if the last line contains only 0s the determinant would be 0 because of the linearity.

[ZioX]

Yes, that works. However, you'd need a proof that you'll always get a row of zeros.

Besides, the proof that the determinant of a diagonal matrix is the product of the diagonal elements is similar enough.

[Icheb]

Alright, thank you very much. :)