Evaluate $\left|\begin{array}{ccc}2 & 3 & -5 \\ 7 & 1 & -2 \\ -3 & 4 & 1\end{array}\right|$ by two methods.
Let $\Delta=\left|\begin{array}{ccc}2 & 3 & -5 \\ 7 & 1 & -2 \\ -3 & 4 & 1\end{array}\right|$
First method
$\Delta=(-1)^{1+1} 2(1+8)+(-1)^{1+2} 3(7-6)+(-1)^{1+3}(-5)(28+3)$
$=2(1+8)-3(7-6)-5(28+3)$
$=18-3-155$
$=-140$
Second method is the Sarus Method, where we adjoin the first two columns to the right to get
$\begin{array}{lllll}2 & 3 & -5 & 2 & 3\end{array}$
$\begin{array}{lllll}7 & 1 & -2 & 7 & 1\end{array}$
$\begin{array}{lllll}-3 & 4 & 1 & -3 & 4\end{array}$
$=(2 \times 1 \times 1+3 \times-2 \times-3-5 \times 7 \times 4)-(-5 \times 1 \times-3+2 \times-2 \times 4+3 \times 7 \times 1)$
$=(2+18-140)-(15-16+21)$
$=-120-20$
$=-140$
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