Why the determinant of A and Row Echelon Form of A is different?
The matrix is \begin{pmatrix}0&-2&-3\\ \:\:\:-1&1&-1\\ \:\:\:2&2&5\end{pmatrix} It's determinant is 6
Reduce to Row Echelon Form: \begin{pmatrix}2&2&5\\ \:\:0&2&\frac{3}{2}\\ \:\:0&0&-\frac{3}{2}\end{pmatrix} The determinant is -6
Why? And the interesting point is why one is 6 and other is -6? Is there any correspondence?
Because when you reduce to row echelon form, you sometimes switch rows which multiplies the determinant by $-1$. If you do it odd number of times, as in your example, you get the $-$ sign in front of the det.