Simple Linear Algebra Proof - Determinants

Question

Prove or disprove the following statement:

If R is the RREF of A, then det A = det R.

So far, I think that this is true, considering A and R are row equivalent, and that the determinant changes as we row reduce in order to compensate for the changing rows. Although, I'm not sure how to go about proving this statement.

Any help would be appreciated, thanks.

Remember that scaling a row also scales the determinant. – Chappers Mar 21 '15 at 17:21 — Chappers, Mar 21 '15 at 17:21
Hint: look at rref($2I$). – David Wheeler Mar 21 '15 at 17:26 — David Wheeler, Mar 21 '15 at 17:26

score 0 · Answer 1 · answered Mar 21 '15 at 17:42

Consider the matrix $$ A =\begin{pmatrix} 2 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 2 \\ \end{pmatrix} $$

The determinant is $\det A = 2 \cdot 2 \cdot 2 = 8$ by multiplying down the diagonal. If you row reduce $A$ to a new matrix $R$ then what is the determinant of $R$?

Simple Linear Algebra Proof - Determinants

1 Answers1