Understanding Determinants
What Is a Determinant?
The determinant is a scalar value computed from a square matrix that encodes important properties. For a 2×2 matrix [[a,b],[c,d]], the determinant is ad − bc. For larger matrices, it is computed by cofactor expansion along any row or column.
Geometric Meaning
The absolute value of the determinant represents the scaling factor of the linear transformation described by the matrix. For a 2×2 matrix, |det| equals the area of the parallelogram formed by the column vectors. For 3×3, it equals the volume of the parallelepiped.
Cofactor Expansion
The determinant is computed by expanding along a row (usually the first): det(A) = Σ (−1)^(i+j) × aᵢⱼ × det(Mᵢⱼ), where Mᵢⱼ is the minor matrix obtained by deleting row i and column j. Each sub-determinant is computed recursively until reaching 2×2 matrices.
Properties of Determinants
If det(A) = 0, the matrix is singular (non-invertible). Swapping two rows changes the sign. Multiplying a row by k multiplies the determinant by k. The determinant of a product equals the product of determinants: det(AB) = det(A)×det(B).
Applications
Determinants are used to check if a matrix is invertible, solve systems of linear equations (Cramer's rule), compute eigenvalues, find areas and volumes, and in multivariable calculus for change of variables in integrals.