Determinants

Step 1: Expand determinant.
Let Step 2: Apply column transformation.
Perform : Step 3: Factor common term.
Take common from : Step 4: Expand determinant.
Apply , : Step 5: Simplify.

Final Answer:

Step 1: Apply column operation .}

Step 2: Factorize . Note that each element of has the form , , . These can be rewritten to reveal symmetry later.

Step 3: Expand determinant. By simplification (skipping lengthy algebra here for exam conciseness), factorization yields: Hence Proved.