Evaluate each determinant.

Key Concepts

Determinant of a Matrix

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). The determinant can also be interpreted geometrically as the volume scaling factor of the linear transformation described 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 a matrix has a row of zeros, its determinant is zero, which indicates that the matrix is singular.

Cofactor Expansion

Cofactor expansion is a method used to calculate 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 corresponding cofactor (which is the determinant of the submatrix formed by removing the row and column of that element), and summing these products. This method is particularly useful for larger matrices.