Question

Find the value of the determinant with out expanding:

$\left|\begin{array}{ccc}5& 2& 3\\ 7& 3& 4\\ 9& 4& 5\end{array}\right|$

Answer 1

Now,

$\left|\begin{array}{ccc}5& 2& 3\\ 7& 3& 4\\ 9& 4& 5\end{array}\right|$

[ $R_2^{\prime }=R_2-R_1$ and $R_3^{\prime }=R_3-R_2$]

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

$=0$. [ Since $R_3=R_2$]