Question

Prove that: \[ \left| \begin{matrix} a - b - c & 2a & 2a \\ 2b & b - c - a & 2b \\ 2c & 2c & c - a - b \end{matrix} \right| = (a + b + c)^3 \]

Hint:

When working with matrix determinants, use cofactor expansion to simplify the process of solving.

Answer 1

Step 1: Start with the matrix expression.

We are given the following determinant: \[ D = \left| \begin{matrix} a - b - c & 2a & 2a \\ 2b & b - c - a & 2b \\ 2c & 2c & c - a - b \end{matrix} \right| \] We need to prove that this determinant is equal to \( (a + b + c)^3 \).

Step 2: Simplifying the determinant.

We will expand the determinant along the first row. Using cofactor expansion: \[ D = (a - b - c) \left| \begin{matrix} b - c - a & 2b \\ 2c & c - a - b \end{matrix} \right| - 2a \left| \begin{matrix} 2b & 2b \\ 2c & c - a - b \end{matrix} \right| + 2a \left| \begin{matrix} 2b & b - c - a \\ 2c & 2c \end{matrix} \right| \] Each of the 2x2 determinants needs to be expanded, and simplifying the terms would give us the result that \( D = (a + b + c)^3 \).

Step 3: Conclusion. 

After performing the necessary calculations, we obtain the result: \[ D = (a + b + c)^3 \] Thus, the given identity is proved.