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Problem 31: Evalaute

\begin{align*} \Delta =\left| \begin{array}{cccc} 1^2& 2^2& 3^2& 4^2\\ 2^2& 3^2& 4^2& 5^2\\ 3^2& 4^2& 5^2& 6^2\\ 4^2& 5^2& 6^2& 7^2\\ \end{array}\right|. \end{align*}

Problem 31: Evalaute

\begin{align*} \Delta =\left| \begin{array}{cccc} 1^2& 2^2& 3^2& 4^2\\ 2^2& 3^2& 4^2& 5^2\\ 3^2& 4^2& 5^2& 6^2\\ 4^2& 5^2& 6^2& 7^2\\ \end{array}\right|. \end{align*}

Solution:

We have

\begin{align*} \Delta =\left| \begin{array}{cccc} 1& 4& 9& 16\\ 4& 9& 16& 25\\ 9& 16& 25& 36\\ 16& 25& 36& 49\\ \end{array}\right| \end{align*} applying $R_4\rightarrow R_4-R_3,R_3\rightarrow R_3-R_2,R_2\rightarrow R_2-R_1$ \begin{align*} \Delta =\left| \begin{array}{cccc} 1& 4& 9& 16\\ 3& 5& 7& 9\\ 5& 7& 9& 11\\ 7& 9& 11& 13\\ \end{array}\right|\\ \end{align*} applying $R_4\rightarrow R_4-R_2,R_3\rightarrow R_3-R_2$ \begin{align*} \Delta =\left| \begin{array}{cccc} 1& 4& 9& 16\\ 3& 5& 7& 9\\ 2& 2& 2& 2\\ 2& 2& 2& 2\\ \end{array}\right|\\ \end{align*} \begin{align*} \Delta =0\\ \end{align*}, the last rows being identical.

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