Technique 1: Row/Column Operations The most powerful technique. Apply -> - or -> - to create zeros. Then expand along the row/column with the most zeros. Remember: adding a multiple of one row to another does NOT change the determinant.
Technique 2: Factor Recognition If a determinant D(a,b,c) becomes 0 when a = b (because two rows become identical), then (a-b) is a factor. Similarly for b = c and a = c. Use this for Vandermonde-type problems.
Technique 3: Sum of Rows/Columns If R1 + R2 + R3 = 0 (or any linear combination of rows gives zero), the determinant is 0. If the sum gives a common factor, use C1 -> C1 + C2 + C3 to factor it out.
Technique 4: Special Determinants Memorize Vandermonde: (a-b)(b-c)(c-a). Recognize circulant determinants. For matrices with entries in AP, the determinant is often 0.
Technique 5: Differentiation For determinants that are functions of x: differentiate one row at a time. Used for "find f'(0)" type problems.
Technique 6: Block Structure For block diagonal or block triangular matrices, det = product of block determinants. Saves massive computation.
Technique 7: Expansion Along Optimal Row/Column Always scan all rows and columns before expanding. Choose the one with the most zeros. If no row has zeros, create them using operations first.