Determinants and Their Properties

GRE Math Subject · Axiom Academy

LESSON Determinants and Their Properties Computing and applying the determinant to understand invertibility and area scaling The determinant of an matrix is the unique function satisfying: Consequence: Swapping two rows multiplies by . Cofactor Expansion: Expanding along row : where is the minor (determinant of the submatrix). Row Reduction: Use row operations: Add multiple of row to another: unchanged Geometric interpretation: is the signed volume scaling factor of the linear transformation represented by . The adjugate (or adjoint) matrix is: Note: This formula is theoretically important but computationally slow for large matrices. Row reduction is preferred in practice. The GRE Subject Test frequently asks: Is a matrix invertible? (compute determinant) Solve systems: is solvable iff Find determinant of composite matrices like or Study tip: Memorize the multiplicative property and the effect of row operations. These are tested most frequently.

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