Evaluate each determinant.

Key Concepts

Determinants in Linear Algebra

A determinant is a scalar value that can be computed from the elements of a square matrix. It provides important information about the matrix, such as whether it is invertible (non-zero determinant) or singular (zero determinant). Determinants can also be used to calculate the area or volume of geometric shapes defined by the matrix.

Properties of Determinants

Determinants have several key properties that simplify their evaluation. For instance, the determinant of a product of matrices equals the product of their determinants, and swapping two rows of a matrix changes the sign of the determinant. Additionally, if two rows (or columns) of a matrix are identical, the determinant is zero.

Cofactor Expansion

Cofactor expansion is a method for calculating the determinant of a matrix by breaking it down into smaller matrices. This technique involves selecting a row or column, multiplying each element by its cofactor (which is the determinant of the submatrix formed by removing the element's row and column), and summing these products. It is particularly useful for larger matrices.