Linear Algebra: Determinants

Definition and Notation

• A determinant is a scalar value that can be computed from the elements of a square matrix.
• Notation: The determinant of a matrix A is denoted as |A| or det(A).

Properties

• Multiplicativity: The determinant of a product of matrices is the product of their determinants: |AB| = |A||B|.
• Additivity: The determinant of a matrix is not additive, but the determinant of a matrix sum can be computed using the formula: |A + B| = |A| + |B| + tr(A^T B) - tr(A)tr(B).
• Scalar multiplication: The determinant of a matrix multiplied by a scalar is the scalar multiplied by the determinant of the matrix: |cA| = c^n |A|, where n is the size of the matrix.

Calculating Determinants

• 2x2 Matrix: The determinant of a 2x2 matrix can be calculated using the formula: |A| = ad - bc, where A = [[a, b], [c, d]].
• NxN Matrix: The determinant of an NxN matrix can be calculated using the formula: |A| = a(ei - fh) - b(di - fg) + c(dh - eg), where A = [[a, b, c], [d, e, f], [g, h, i]].
• Laplace Expansion: The determinant of an NxN matrix can be calculated using the Laplace expansion, which involves expanding the determinant along a row or column and summing the determinants of the resulting sub-matrices.

Applications

• Linear Independence: A set of vectors is linearly independent if and only if the determinant of the matrix formed by the vectors is non-zero.
• Invertibility: A matrix is invertible if and only if its determinant is non-zero.
• Volume and Area: The determinant of a matrix can be used to calculate the volume and area of a parallelepiped and a parallelogram, respectively.

Important Theorems

• Cramer's Rule: The solution to a system of linear equations can be expressed using determinants, where the determinant of the coefficient matrix is used to find the denominators of the solution.
• Inverse Matrix: The inverse of a matrix can be expressed using determinants, where the determinant of the matrix is used to find the inverse.