2x2 Determinant Formula: Linear Algebra Determinants Quiz and Flashcards

Study Notes

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.

Description

Learn about the definition, properties, and calculations of determinants in linear algebra, including its applications in linear independence, invertibility, and volume/area calculations. Practice your skills with this quiz!