Using properties of determinants, show that Using Properties of determinants, show that:
$$
\begin{vmatrix}
a &   a+b & a+2b\\
a+2b & a  &  a+b\\
a+b & a+2b &  a
\end{vmatrix}
 =  9b^2 (a+b)
$$
I've tried it but not getting $9b^2$

Add all the three columns to get
$$\left \vert \begin{bmatrix} 3a+3b & a+b & a+2b\\ 3a+3b & a & a+b\\ 3a+3b & a+2b & a\end{bmatrix}\right \vert = 3(a+b)\left \vert \begin{bmatrix} 1 & a+b & a+2b\\ 1 & a & a+b\\ 1 & a+2b & a\end{bmatrix}\right \vert$$
Subtract first row from row and first from third to get
$$3(a+b)\left \vert \begin{bmatrix} 1 & a+b & a+2b\\ 1 & a & a+b\\ 1 & a+2b & a\end{bmatrix}\right \vert = 3(a+b)\left \vert \begin{bmatrix} 1 & a+b & a+2b\\ 0 & -b & -b\\ 0 & b & -2b\end{bmatrix}\right \vert = 3(a+b)(2b^2+b^2) = 9b^2(a+b)$$
