Verify Property $2$ for $\Delta=\left|\begin{array}{ccc}2 & -3 & 5 \\ 6 & 0 & 4 \\ 1 & 5 & -7\end{array}\right|$
(N/A) The determinant is given by $\Delta = \left| \begin{array}{ccc} 2 & -3 & 5 \\ 6 & 0 & 4 \\ 1 & 5 & -7 \end{array} \right|$.
Expanding along the first row:
$\Delta = 2(0 - 20) - (-3)(-42 - 4) + 5(30 - 0) = 2(-20) + 3(-46) + 5(30) = -40 - 138 + 150 = -28$.
Interchanging rows $R_{2}$ and $R_{3}$ i.e.,$R_{2} \leftrightarrow R_{3}$,we get $\Delta_{1} = \left| \begin{array}{ccc} 2 & -3 & 5 \\ 1 & 5 & -7 \\ 6 & 0 & 4 \end{array} \right|$.
Expanding $\Delta_{1}$ along the first row:
$\Delta_{1} = 2(20 - 0) - (-3)(4 + 42) + 5(0 - 30) = 2(20) + 3(46) + 5(-30) = 40 + 138 - 150 = 28$.
Clearly,$\Delta_{1} = -\Delta$,which is $28 = -(-28)$.
Hence,Property $2$ is verified.
Expanding along the first row:
$\Delta = 2(0 - 20) - (-3)(-42 - 4) + 5(30 - 0) = 2(-20) + 3(-46) + 5(30) = -40 - 138 + 150 = -28$.
Interchanging rows $R_{2}$ and $R_{3}$ i.e.,$R_{2} \leftrightarrow R_{3}$,we get $\Delta_{1} = \left| \begin{array}{ccc} 2 & -3 & 5 \\ 1 & 5 & -7 \\ 6 & 0 & 4 \end{array} \right|$.
Expanding $\Delta_{1}$ along the first row:
$\Delta_{1} = 2(20 - 0) - (-3)(4 + 42) + 5(0 - 30) = 2(20) + 3(46) + 5(-30) = 40 + 138 - 150 = 28$.
Clearly,$\Delta_{1} = -\Delta$,which is $28 = -(-28)$.
Hence,Property $2$ is verified.
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