Using the property of determinants and without expanding, prove that:
Question:

Using the property of determinants and without expanding, prove that:

$\left|\begin{array}{lll}2 & 7 & 65 \\ 3 & 8 & 75 \\ 5 & 9 & 86\end{array}\right|=0$

Solution:

$\left|\begin{array}{lll}2 & 7 & 65 \\ 3 & 8 & 75 \\ 5 & 9 & 86\end{array}\right|=\left|\begin{array}{lll}2 & 1 & 65+2 \\ 3 & 8 & 72+3 \\ 5 & 9 & 81+5\end{array}\right|$

$=\left|\begin{array}{lll}2 & 7 & 63 \\ 3 & 8 & 72 \\ 5 & 9 & 81\end{array}\right|+\left|\begin{array}{lll}2 & 7 & 2 \\ 3 & 8 & 3 \\ 5 & 9 & 5\end{array}\right|$

$=\left|\begin{array}{lll}2 & 7 & 9(7) \\ 3 & 8 & 9(8) \\ 5 & 9 & 9(9)\end{array}\right|+0 \quad$ [Two columns are identical]

$=9\left|\begin{array}{lll}2 & 7 & 7 \\ 3 & 8 & 8 \\ 5 & 9 & 9\end{array}\right|$

$=0$           [Two columns are identical]