Question

The value of $\left|\begin{array}{ccc}(a+d)(a+2d)& a+2d& a\\ 2d(a+2d)& d& d\\ 2d(a+3d)& d& d\end{array}\right|$ is

• A. $4d$
• B. $4d^2$
• C. $4d^3$
• D. $4d^4$

Answer 1

Answer: (D)

$4d^4$

$\left|\begin{array}{ccc}(a+d)(a+2d)& a+2d& a\\ 2d(a+2d)& d& d\\ 2d(a+3d)& d& d\end{array}\right|$
$=d^2\left|\begin{array}{ccc}(a+d)(a+2d)& a+2d& a\\ 2(a+2d)& 1& 1\\ 2(a+3d)& 1& 1\end{array}\right|$
$R_2{\to R}_2{-R}_3$
$=d^2\left|\begin{array}{ccc}(a+d)(a+2d)& a+2d& a\\ -2d& 0& 0\\ 2(a+3d)& 1& 1\end{array}\right|$
$=d^2\left[2d\right(a+2d-a\left)\right]={4d}^4$