Determinants: Properties and Calculation

What is a Determinant

A determinant is a scalar associated with a square matrix. The intuition: det A is the “volume” of the parallelepiped formed by the rows (or columns) of the matrix. If det A = 0, the columns are linearly dependent—the “parallelepiped” is degenerate (flat).

For 2×2: det[[a,b],[c,d]] = ad − bc.

Geometrically: |ad−bc| is the area of the parallelogram with sides (a,b) and (c,d).

The Axiomatic Definition of the Determinant

The determinant is the unique function of the rows of a matrix possessing three properties:

1. Multilinearity in the rows
2. Skew-symmetry (swapping two rows changes the sign)
3. det E = 1

Expansion Along a Row/Column

det A = Σⱼ aᵢⱼ Aᵢⱼ, where Aᵢⱼ = (−1)^(i+j) Mᵢⱼ is the cofactor, and Mᵢⱼ is the minor (the determinant of the submatrix without the i-th row and j-th column).

For 3×3 along the first row: det A = a₁₁(a₂₂a₃₃−a₂₃a₃₂) − a₁₂(a₂₁a₃₃−a₂₃a₃₁) + a₁₃(a₂₁a₃₂−a₂₂a₃₁).

Properties of the Determinant

• det Aᵀ = det A
• det(AB) = det A · det B
• det(A⁻¹) = 1/det A
• If a row is a linear combination of other rows, then det = 0

Elementary Transformations:

• Swapping two rows: det changes sign
• Adding a multiple of one row to another: det does not change
• Multiplying a row by λ: det is multiplied by λ

Cramer's Formula

The system Ax = b (square, det A ≠ 0) has a unique solution xᵢ = det Aᵢ / det A, where Aᵢ is the matrix with the i-th column replaced by b.

Application: important theoretically, but computationally inefficient—Gaussian elimination works faster.

Geometric Applications

Volume of a parallelepiped: V = |det[a,b,c]| (rows are the vectors of the edges).

Orientation: sign(det) determines the orientation of the basis.

Jacobian matrix: when changing variables in an integral, the volume stretching coefficient = |det J|.