Evaluate determinant along second column | ccc 1 & -1 & 2 3 & 2 &… - Vidyadip

Question

Evaluate determinant along second column $\left|\begin{array}{ccc}1 & -1 & 2 \\ 3 & 2 & -2 \\ 0 & 1 & -2\end{array}\right|$

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Answer

Here, $\left|\begin{array}{lll}a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33}\end{array}\right|=\left|\begin{array}{rrr}1 & -1 & 2 \\ 3 & 2 & -2 \\ 0 & 1 & -2\end{array}\right|$

Expansion along the second column

$\begin{aligned} & =a_{12} C_{12}+a_{22} C_{22}+a_{32} C_{32} \\ & =(-1) \times(-1)^{1+2}\left|\begin{array}{rr}3 & -2 \\ 0 & -2\end{array}\right| \\ & +(2) \times(-1)^{2+2}\left|\begin{array}{rr}1 & 2 \\ 0 & -2\end{array}\right|+(1) \times(-1)^{3+2}\left|\begin{array}{rr}1 & 2 \\ 3 & -2\end{array}\right|\end{aligned}$

$\begin{aligned} & =1(-6-0)+2(-2-0)-1(-2-6) \\ & =1(-6)+2(-2)-1(-8)\end{aligned}$

$\begin{aligned} & =-6-4+8 \\ & =-2\end{aligned}$

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