Question

The value of $ \left| \begin{matrix} {{1}^{2}} & {{2}^{2}} & {{3}^{2}} \\ {{2}^{2}} & {{3}^{2}} & {{4}^{2}} \\ {{3}^{2}} & {{4}^{2}} & {{5}^{2}} \\ \end{matrix} \right| $ is:

• A. 8
• B. $ -8 $
• C. 400
• D. 1
• E. 0

Answer 1

Answer: (B)

$ \left| \begin{matrix} {{1}^{2}} & {{2}^{2}} & {{3}^{2}} \\ {{2}^{2}} & {{3}^{2}} & {{4}^{2}} \\ {{3}^{2}} & {{4}^{2}} & {{5}^{2}} \\ \end{matrix} \right| $ $ =\left| \begin{matrix} 1 & 4 & 9 \\ 4 & 9 & 16 \\ 9 & 16 & 25 \\ \end{matrix} \right|=\left| \begin{matrix} 1 & 3 & 5 \\ 4 & 5 & 7 \\ 9 & 7 & 9 \\ \end{matrix} \right| $ $ {{C}_{2}}\to {{C}_{2}}-{{C}_{1}}and\,{{C}_{3}}\to {{C}_{3}}-{{C}_{2}} $ $ =\left| \begin{matrix} 1 & 3 & 2 \\ 4 & 5 & 2 \\ 9 & 7 & 2 \\ \end{matrix} \right| $ $ {{C}_{3}}\to {{C}_{3}}-{{C}_{2}} $ $ =\left| \begin{matrix} 1 & 3 & 2 \\ 3 & 2 & 0 \\ 5 & 2 & 0 \\ \end{matrix} \right| $ $ \begin{align} & {{R}_{2}}\to {{R}_{2}}-{{R}_{1}} \\ & {{R}_{3}}\to {{R}_{3}}-{{R}_{2}} \\ \end{align} $ $ =2\left| \begin{matrix} 3 & 2 \\ 5 & 2 \\ \end{matrix} \right|=2(6-10)=-8 $