Show that $$ \left|\begin{array}{lll} a_{1} & b_{1} & c_{1} \\ a_{2} & b_{2} & c_{2} \\ a_{3} & b_{3} & c_{3} \end{array}\right|=\left|\begin{array}{ccc} a_{1}+k b_{1} & b_{1} & c_{1} \\ a_{2}+k b_{2} & b_{2} & c_{2} \\ a_{3}+k b_{3} & b_{3} & c_{3} \end{array}\right| $$
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Show that $$\left|\begin{array}{lll} a_{1} & b_{1} & c_{1} \\ a_{2} & b_{2} & c_{2} \\ a_{3} & b_{3} & c_{3} \end{array}\right|=\begin{array}{l} =a_{1} b_{2} c_{3}+b_{1} c_{2} a_{3}+c_{1} a_{2} b_{3} \\ -a_{3} b_{2} c_{1}-b_{3} c_{2} a_{1}-b_{1} a_{2} c_{3} \end{array}$$ by expanding down the second column.
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