Solve this

Solution:

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

$\begin{array}{lllll}2 & 3 & -5 & 2 & 3\end{array}$

$\begin{array}{lllll}7 & 1 & -2 & 7 & 1\end{array}$

$\begin{array}{lllll}-3 & 4 & 1 & -3 & 4\end{array}$

$=(2 \times 1 \times 1+3 \times-2 \times-3-5 \times 7 \times 4)-(-5 \times 1 \times-3+2 \times-2 \times 4+3 \times 7 \times 1)$

$=(2+18-140)-(15-16+21)$

$=-120-20$

 

$=-140$