Given a square matrix A, is the determinant of A equals to the product of the diagonal elements of A after the echelon?

I know that making operations with lines change the determinant, but I did it in some matrices and all of them had the same determinant than the product of the diagonal elements of itself after echeloning.

Is this proprierty true?

Can someone give me a hint on how to prove it, if it's true?

For simplicity, let A non singular.

Thank you!

  • $\begingroup$ If you're saying what I think you're saying, then the answer is yes as long as you don't multiply a row by a constant and don't switch the rows. $\endgroup$ – Omnomnomnom Nov 19 '15 at 3:56
  • $\begingroup$ See the answer to this question for the operations you're allowed to do on determinants and how they affect the value of the determinant. $\endgroup$ – user137731 Nov 19 '15 at 3:58

Here's a set of information to help you answer yourself:

  • A matrix's determinant is unaffected by an operation of the form: $L_i \to L_i + \alpha L_j$.

  • If $B$ is obtained from $A$ by interchanging two rows, then $\det B = - \det A$

  • If $B$ is obtained from $A$ by an operation: $L_i \to \alpha L_i$, then $\det B = \alpha \det A$

  • The determinant of a triangular matrix is the product of its diagonal entries.

  • $\begingroup$ thank you very much, I didnt remember to use this rules. $\endgroup$ – DcCoO Nov 19 '15 at 4:00

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