Strategy 1: Row/Column Operations First Before expanding, simplify using $R_i$ -> $R_i$ - $R_j$ or $C_i$ -> $C_i$ - $C_j$ to create zeros. Target: at least 2 zeros in one row/column.

Strategy 2: Factor Recognition If the determinant has a common factor in a row/column, pull it out. If substituting a = b makes two rows identical, then (a-b) is a factor.

Strategy 3: Property-Based Shortcuts

• Two identical rows => det = 0 (no computation needed)
• Triangular matrix => product of diagonal
• If all row sums equal k, then [1,1,1]^T is an eigenvector with eigenvalue k, meaning det contains k as a factor

Strategy 4: Differentiation/Integration For determinants that are functions of x: differentiate row by row, or evaluate at special values (x=0, x=1) to identify the polynomial.

Strategy 5: Summation of Determinants If asked for a sum of determinants, check if they can be combined into a single determinant by using the linearity property along one row.