Question

The value of the determinant $\left|\begin{array}{ccc}x+1& x+2& x+4\\ x+3& x+5& x+8\\ x+7& x+10& x+14\end{array}\right|$ is

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

Answer 1

The correct option is A $-2$

Let $\Delta =\left|\begin{array}{ccc}x+1& x+2& x+4\\ x+3& x+5& x+8\\ x+7& x+10& x+14\end{array}\right|$

Applying $C_3\to C_3-C_2$

$\Delta =\left|\begin{array}{ccc}x+1& x+2& 2\\ x+3& x+5& 3\\ x+7& x+10& 4\end{array}\right|$

Applying $C_2\to C_2-C_1$

$\Delta =\left|\begin{array}{ccc}x+1& 1& 2\\ x+3& 2& 3\\ x+7& 3& 4\end{array}\right|$

Applying $R_2\to R_2-R_1,R_3\to R_3-R_2$

$\Delta =\left|\begin{array}{ccc}x+1& 1& 2\\ 2& 1& 1\\ 4& 1& 1\end{array}\right|$

Applying $R_3\to R_3-R_2$

$\Delta =\left|\begin{array}{ccc}x+1& 1& 1\\ 2& 1& 0\\ 2& 0& 0\end{array}\right|=2(0-1)=-2$