In Exercises 46–51, evaluate each determinant.

Key Concepts

Determinants

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 and the volume scaling factor of the linear transformation represented by the matrix. The determinant can be calculated using various methods, including cofactor expansion and row reduction.

Square Matrices

A square matrix is a matrix with the same number of rows and columns, denoted as n x n. Determinants are only defined for square matrices, as they represent linear transformations in n-dimensional space. Understanding the properties of square matrices, such as rank and eigenvalues, is essential for evaluating their determinants.

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. It is particularly useful for larger matrices.