Strategic expansion: Always expand along the row or column with the most zeros. Before expanding, use row/column operations to CREATE zeros.

Example: For a 3x3 determinant, if you can make two entries in a row zero using $R_i$ -> $R_i$ - $kR_j$ operations, then expansion along that row requires computing only ONE 2x2 determinant.

Cofactor identity trap: The sum of products of elements of any row with cofactors of a DIFFERENT row equals zero. That is: a11A21 + a12A22 + a13*A23 = 0 (elements of R1 with cofactors of R2). This property is tested directly in JEE.

Block matrix determinant: For block diagonal or block triangular matrices: det = det(block1) * det(block2). Saves massive computation for structured matrices.