# Proof strategy about a property of triangular matrices

Is it by mathematical induction the best way to prove that the determinant of an upper (lower) triangular matrix is the product of the elements of the main diagonal? Actually, I am wondering about which is the most intuitive strategy for proving this claim.

• You can also prove it directly from the definition of the determinant as a sum of products over permutations by noting that the only permutation contributing a non-zero product is the identity permutation. I consider this also quite intuitive. – levap Nov 16 '15 at 16:49
• This is a forum for questions and answers that are closed form....starting a question with "In your opinion" is practically begging for it to be closed as "opinion based" – Alan Nov 16 '15 at 16:49
• @Ievap, I agree. I like more your formulation. Perhaps, could you expand it more in an answer, please? – Always learning Nov 16 '15 at 16:51
• @Alan, thanks...I have improved my question. – Always learning Nov 16 '15 at 16:53

## 1 Answer

I find induction and Laplace expansion on the first column or row to be the easiest way.

• do you mean them as used together or as 2 different ways to prove the statement? – Always learning Nov 16 '15 at 18:32
• @Alwayslearning, together. – lhf Nov 16 '15 at 18:37