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 be calculated using various methods, including row reduction, cofactor expansion, or specific formulas for small matrices.

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 are identical, the determinant is zero, which indicates linear dependence among the rows.

Applications of Determinants

Determinants are used in various applications, including solving systems of linear equations, finding the area or volume of geometric shapes, and analyzing the stability of systems in engineering. They also play a crucial role in calculus, particularly in the context of Jacobians when changing variables in multiple integrals.