Question

Show that $\left|\begin{array}{ccc}a& a+b& a+2b\\ a+2b& a& a+b\\ a+b& a+2b& a\end{array}\right|=9b^2(a+b)$.

Answer 1

Take L.H.S

$\left[\begin{array}{ccc}a& a+b& a+2b\\ a+2b& a& a+b\\ a+b& a+2b& a\end{array}\right]$

Now apply $C_1\to C_1+C_2+C_3$

$\left[\begin{array}{ccc}3a+3b& a+b& a+2b\\ 3a+3b& a& a+b\\ 3a+3b& a+2b& a\end{array}\right]$

Taking common $3(a+b)$ from $C_1$

$3(a+b)\left[\begin{array}{ccc}1& a+b& a+2b\\ 1& a& a+b\\ 1& a+2b& a\end{array}\right]3(a+b)$

Now apply, $R_1\to R_1-R_3$

$R_2\to R_2-R_3$

$3(a+b)\left[\begin{array}{ccc}0& -b& 2b\\ 0& -2b& a+b\\ 1& a+2b& a\end{array}\right]3(a+b)$

$3(a+b)\left[\begin{array}{cc}-b& 2b\\ -2b& b\end{array}\right]3(a+b)$

$3b^2(a+b)\left[\begin{array}{cc}-1& 2\\ 2& 1\end{array}\right]$

$3b^2(a+b)(-1+4)=9b^2(a+b)$

Hence proved.