Question

Using the property of determinants and without expanding, prove that: $\begin{vmatrix}2&7&65\\\3&8&75\\\5&9&86\end{vmatrix}$=0

Answer 1

$\begin{vmatrix}2&7&65\\\3&8&75\\\5&9&86\end{vmatrix}$

=$\begin{vmatrix}2&7&63+2\\\3&8&72+3\\\5&9&81+5\end{vmatrix}$

=$\begin{vmatrix}2&7&63\\\3&8&72\\\5&9&81\end{vmatrix}+\begin{vmatrix}2&7&2\\\3&8&3\\\5&9&5\end{vmatrix}$

=$\begin{vmatrix}2&7&9(7)\\\3&8&9(8)\\\5&9&9(9)\end{vmatrix}+0$ [Two columns are identical]

=$9\begin{vmatrix}2&7&7\\\3&8&8\\\5&9&9\end{vmatrix}$ [Two columns are identical]

=0