Question

Calculate the values of the determinants:
$\left|\begin{array}{cccc}1& 1& 1& 1\\ 1& 2& 3& 4\\ 1& 3& 6& 10\\ 1& 4& 10& 20\end{array}\right|$.

Answer 1

$\Delta =\left|\begin{array}{cccc}1& 1& 1& 1\\ 1& 2& 3& 4\\ 1& 3& 6& 10\\ 1& 4& 10& 20\end{array}\right|C_4\to C_4-C_3\Rightarrow \Delta =\left|\begin{array}{cccc}1& 1& 1& 0\\ 1& 2& 3& 1\\ 1& 3& 6& 4\\ 1& 4& 10& 10\end{array}\right|C_3\to C_3-C_2\Rightarrow \Delta =\left|\begin{array}{cccc}1& 1& 0& 0\\ 1& 2& 1& 1\\ 1& 3& 3& 4\\ 1& 4& 6& 10\end{array}\right|C_2\to C_2-C_1\Rightarrow \Delta =\left|\begin{array}{cccc}1& 0& 0& 0\\ 1& 1& 1& 1\\ 1& 2& 3& 4\\ 1& 3& 6& 10\end{array}\right|$

Expanding along $R_1$

$\Rightarrow \Delta =1\left|\begin{array}{ccc}1& 1& 1\\ 2& 3& 4\\ 3& 6& 10\end{array}\right|-0+0-0\Rightarrow \Delta =\left|\begin{array}{ccc}1& 1& 1\\ 2& 3& 4\\ 3& 6& 10\end{array}\right|C_3\to C_3-C_2\Rightarrow \Delta =\left|\begin{array}{ccc}1& 1& 0\\ 2& 3& 1\\ 3& 6& 4\end{array}\right|C_2\to C_2-C_1\Rightarrow \Delta =\left|\begin{array}{ccc}1& 0& 0\\ 2& 1& 1\\ 3& 3& 4\end{array}\right|$

Expanding along $R_1$

$\Rightarrow \Delta =1\{4-3\}-0+0\Rightarrow \Delta =1$