Question

Prove the following using properties of determinants :

\[\begin{vmatrix}a + b + 2c & a & b \\ c & b + c + 2a & b \\ c & a & c + a + 2b\end{vmatrix} = 2\left( a + b + c \right) {}^3\]

Answer 1

Let \[\bigtriangleup = \begin{vmatrix}a + b + 2c & a & b \\ c & b + c + 2a & b \\ c & a & c + a + 2b\end{vmatrix}\]

Applying  \[C_1 \to C_1 + C_2 + C_3\] , we get: 

\[\bigtriangleup = \begin{vmatrix}2a + 2b + 2c & a & b \\ 2a + 2b + 2c & b + c + 2a & b \\ 2a + 2b + 2c & a & c + a + 2b\end{vmatrix}\]

\[\Rightarrow \bigtriangleup = 2\left( a + b + c \right)\begin{vmatrix}1 & a & b \\ 1 & b + c + 2a & b \\ 1 & a & c + a + 2b\end{vmatrix}\]

Now, applying

\[R_2 \to R_2 - R_1 \text { and } R_3 \to R_3 - R_1\] , we get:

\[\Rightarrow \bigtriangleup = 2\left( a + b + c \right)\begin{vmatrix}1 & a & b \\ 0 & b + c + a & 0 \\ 0 & 0 & c + a + b\end{vmatrix}\]

\[\Rightarrow \bigtriangleup = 2 \left( a + b + c \right)^3\]

∴ \[\begin{vmatrix}a + b + 2c & a & b \\ c & b + c + 2a & b \\ c & a & c + a + 2b\end{vmatrix} = 2\left( a + b + c \right) {}^3\]