Question

Verify property $2$ for
$\Delta =\left|\begin{array}{ccc}2& -3& 5\\ 6& 0& 4\\ 1& 5& -7\end{array}\right|$
Property $2:$ If any two rows (or columns) of a determinant are interchanged, then sign of determinant changes.

Answer 1

Given: $\Delta =\left|\begin{array}{ccc}2& -3& 5\\ 6& 0& 4\\ 1& 5& -7\end{array}\right|$
Expanding the determinants along first row,
$\Delta =2\left|\begin{array}{cc}0& 4\\ 5& -7\end{array}\right|-(-3)\left|\begin{array}{cc}6& 4\\ 1& -7\end{array}\right|+5\left|\begin{array}{cc}6& 0\\ 1& 5\end{array}\right|$
$=2\left(0\right(7)-5(4\left)\right)+3\left(6\right(7)-(1\left)\right(4\left)\right)+5\left(6\right(5)-1(0\left)\right)$
$=2(0-20)+3(-42-4)+5(30-0)$
$=2(-20)+3(-46)+5\left(30\right)$
$=-40-138+150$
$=-28$

Interchanging $R_2$ and $R_3$ i.e., $R_2\leftrightarrow R_3$, We get
${\Delta }_1=\left|\begin{array}{ccc}2& -3& 5\\ 1& 5& -7\\ 6& 0& 4\end{array}\right|$
$=2\left|\begin{array}{cc}5& -7\\ 0& 4\end{array}\right|-(-3)\left|\begin{array}{cc}1& -7\\ 6& 4\end{array}\right|+5\left|\begin{array}{cc}1& 5\\ 6& 0\end{array}\right|$
$=2\left(5\right(4)-0(-7\left)\right)+3\left(1\right(4)-6(-7\left)\right)+5\left(1\right(0)-6(5\left)\right)$
$=2(20-0)+3(4+42)+5(0-30)$
$=2\left(20\right)+3\left(46\right)+5(-30)$
$=40+138-150$
$=28$
${\Delta }_1=28$
$=-(-28)$
$=-\Delta$
Therefore, ${\Delta }_1=-\Delta$
Hence property $2^{nd}$ is verified.
i.e., If any two rows (or columns) of a determinant are interchanged, then sign of determinant changes.