Question

The values of the determinant $\begin{vmatrix} x+1& x+2 &x+4 \\\[0.3em] x+3 & x+5 & x+8 & \\\[0.3em] x+7 & x+10 & x+14 \end{vmatrix}$

• A. -2
• B. $x^2 + 2$
• C. 2
• D. none of these

Answer 1

Answer: (A)

-2

Answer 2

Operate $R_2 - R_1; R_3 - R_1$ we get the determinant as $\begin{vmatrix}x+1 &x+2&x+4\\\ 2& 3 &4 \\\ 6 &8 &10\end{vmatrix}$ operate $C_3 - C_2; C_2 - C_1$, we get the value = $\begin{vmatrix}x+1 &1&3\\\ 2& 1 &2 \\\ 6 &2 &4\end{vmatrix}$ operate $R_3 - 2 \,R_2$, we get $\begin{vmatrix}x+1 &1&3\\\ 2& 1 &2 \\\ 2 &0 &0\end{vmatrix} = 2 \begin{vmatrix}1&3\\\ 1&2\end{vmatrix} = 2 ( 2 - 3) = - 2$