A) 0
B) \[-1\]
C) 1
D) 10
Correct Answer: C
Solution :
We have \[\left| \begin{matrix} 1 & 1 & 1 & 1 \\ 1 & 2 & 3 & 4 \\ 1 & 3 & 6 & 10 \\ 1 & 4 & 10 & 20 \\ \end{matrix} \right|=\left| \begin{matrix} 1 & 1 & 1 & 1 \\ 0 & 1 & 2 & 3 \\ 0 & 2 & 5 & 9 \\ 0 & 3 & 9 & 10 \\ \end{matrix} \right|\] (applying\[{{R}_{2}}\to {{R}_{2}}-{{R}_{1}},{{R}_{3}}\to {{R}_{3}}-{{R}_{1}}\]and\[{{R}_{4}}\to {{R}_{4}}-{{R}_{1}})\] \[=\left| \begin{matrix} 1 & 2 & 3 \\ 2 & 5 & 9 \\ 3 & 9 & 19 \\ \end{matrix} \right|=\left| \begin{matrix} 1 & 2 & 3 \\ 0 & 1 & 3 \\ 0 & 3 & 10 \\ \end{matrix} \right|\] (applying\[{{R}_{2}}\to {{R}_{2}}-2{{R}_{1}},{{R}_{3}}\to {{R}_{3}}-3{{R}_{1}}\])\[=(10-9)=1\]
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