Question

The determinant $\left|\begin{array}{ccc}a& a+d& a+2d\\ a^2& {(a+d)}^2& {(a+2d)}^2\\ 2a+3d& 2(a+d)& 2a+d\end{array}\right|=0.$ Then

• A. $d=0$
• B. $a+d=0$
• C. $d=0$ or $a+d=0$
• D. none of these

Answer 1

The correct option is C $d=0$ or $a+d=0$

$\left|\begin{array}{ccc}a& a+d& a+2d\\ a^2& {(a+d)}^2& {(a+2d)}^2\\ 2a+3d& 2(a+d)& 2a+d\end{array}\right|=0$
$(a+d)\left|\begin{array}{ccc}a& 1& a+2d\\ a^2& a+d& {(a+2d)}^2\\ 2a+3d& 2& 2a+d\end{array}\right|=0$
$R_3\to R_3-2R_1$
$(a+d)\left|\begin{array}{ccc}a& 1& a+2d\\ a^2& a+d& {(a+2d)}^2\\ 3d& 0& -3d\end{array}\right|=0$
$3d(a+d)\left|\begin{array}{ccc}a& 1& a+2d\\ a^2& a+d& {(a+2d)}^2\\ 1& 0& -1\end{array}\right|=0$
$C_1\to C_1+C_3$
$3d(a+d)\left|\begin{array}{ccc}2a+2d& 1& a+2d\\ a^2+{(a+2d)}^2& a+d& {(a+2d)}^2\\ 0& 0& -1\end{array}\right|=0$
$-3d(a+d)[2{(a+d)}^2-a^2-{(a+2d)}^2]=0$
$\Rightarrow 6d^3(a+d)=0$
$\Rightarrow d=0ora+d=0$