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Question
Evaluate
\[\begin{vmatrix}2 & 3 & - 5 \\ 7 & 1 & - 2 \\ - 3 & 4 & 1\end{vmatrix}\] by two methods.
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Solution
Let
\[∆ = \begin{vmatrix}2 & 3 & - 5 \\ 7 & 1 & - 2 \\ - 3 & 4 & 1\end{vmatrix}\]
First method
\[∆ = \left( - 1 \right)^{1 + 1} 2\left( 1 + 8 \right) + \left( - 1 \right)^{1 + 2} 3\left( 7 - 6 \right) + \left( - 1 \right)^{1 + 3} \left( - 5 \right)\left( 28 + 3 \right)\]
\[ = 2\left( 1 + 8 \right) - 3\left( 7 - 6 \right) - 5\left( 28 + 3 \right)\]
\[ = 18 - 3 - 155\]
\[ = - 140\]
Second method is the Sarus Method, where we adjoin the first two columns to the right to get
`|[2,3,-5,2,3],[7,1,-2,7,1],[-3,4,1,-3,4]|`
=(2 x 1 x 1 + 3 x -2 x -3 -5 x 7 x 4)-(-5 x 1 x -3 + 2 x -2 x 4 + 3 x 7 x 1)
=(2 + 18 - 140) - (15 - 16 + 21)
= -120 - 20
= -140
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