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Problem 26: Find the value of determinant along $R_2$

Problem 26: Find the value of determinant

\begin{align*} \Delta =\left|\begin{array}{ccc}1& 2& 7\\ 5& 0& 2\\ 3& -4& 6\end{array}\right| \end{align*}

Solution:

Expanding $\Delta$ along the $R_2$ as maximum zeros are in $R_2$, we get

\begin{align*} \Delta = (-1)^{2+1}5\left|\begin{array}{cc}2& 7\\ -4& 6\end{array}\right| (-1)^{2+2}0\left|\begin{array}{cc}1& 7\\ 3& 6\end{array}\right| (-1)^{2+3}2\left|\begin{array}{cc}1& 2\\ 3& -4\end{array}\right|\\ \Delta = -5(12+28)+0-2(-4-6)=-200+20=-180. \end{align*}

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