Question

Find the value of the determinant $\left|\begin{array}{ccc}1& 1& 1\\ 15& 20& 25\\ 32& 44& 64\end{array}\right|$ is :

Answer 1

$\left|\begin{array}{ccc}1& 1& 1\\ 15& 20& 25\\ 32& 44& 64\end{array}\right|$

$R_3\to \frac{R_3}{4}$ and $R_2\to \frac{R_2}{5}$

$=20\left|\begin{array}{ccc}1& 1& 1\\ 3& 4& 5\\ 8& 11& 16\end{array}\right|$

$R_3\to R_3-R_2$

$=20\left|\begin{array}{ccc}1& 1& 1\\ 3& 4& 5\\ 5& 7& 11\end{array}\right|$

$R_3\to R_3-5R_1$

$=20\left|\begin{array}{ccc}1& 1& 1\\ 3& 4& 5\\ 0& 2& 6\end{array}\right|$

$R_3\to \frac{R_3}{2}$

$=40\left|\begin{array}{ccc}1& 1& 1\\ 3& 4& 5\\ 0& 1& 3\end{array}\right|$

$R_2\to R_2-3R_1$

$=40\left|\begin{array}{ccc}1& 1& 1\\ 0& 1& 2\\ 0& 1& 3\end{array}\right|$

$=40[1(3-2)-1(0-0)+1(0-0)]$

$=40\times 1=40$