There's a property that says when you interchange two rows/columns from a matrix A, the resulting determinant B will have its determinant equal to the original one, but with its sign inversed: $\det(B)=-\det(A)$. I've been wanting to know how to explain it for this specific case (without using elementary matrix):
For the A matrix (the original matrix) I've chosen to do cofactor expasion on the first row: $$ |A| =\ \ \ \begin{vmatrix} a_{11}& a_{12}& a_{13}\\ a_{21}& a_{22}& a_{23}\\ a_{31}& a_{32}& a_{33}\\ \end{vmatrix} = a_{11} \begin{vmatrix} a_{22}& a_{23}\\ a_{32}& a_{33}\\ \end{vmatrix} -a_{12} \begin{vmatrix} a_{21}& a_{23}\\ a_{31}& a_{33}\\ \end{vmatrix} +a_{13} \begin{vmatrix} a_{21}& a_{22}\\ a_{31}& a_{32}\\ \end{vmatrix} $$ Now, when I interchange the first row of the A matrix with its second row, I'll have this (for which I've chosen to do the row expansion on the second row, since it's the same row as the one I used for A): $$ |B| =\ \ \ \begin{vmatrix} a_{21}& a_{22}& a_{23}\\ a_{11}& a_{12}& a_{13}\\ a_{31}& a_{32}& a_{33}\\ \end{vmatrix} = a_{11} \begin{vmatrix} a_{22}& a_{23}\\ a_{32}& a_{33}\\ \end{vmatrix} -a_{12} \begin{vmatrix} a_{21}& a_{23}\\ a_{31}& a_{33}\\ \end{vmatrix} +a_{13} \begin{vmatrix} a_{21}& a_{22}\\ a_{31}& a_{32}\\ \end{vmatrix} $$ So, it clearly sounds like |A| = |B| by looking at these, although i'm perfectly sure it's not. What am I missing here? =\
You've done the cofactor expansion incorrectly. When you expand along the middle row for $B$, you start with $-$ first, not $+$.