Use the determinant theorems to evaluate each determinant. See Example 4.

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 (non-zero determinant) or singular (zero determinant). Determinants can be calculated using various methods, including expansion by minors, row reduction, or specific formulas for small matrices.

Determinant Theorems

Determinant theorems are rules and properties that simplify the calculation of determinants. Key theorems include the properties of linearity, the effect of row operations on determinants, and the relationship between determinants and matrix inverses. Understanding these theorems allows for more efficient evaluation of determinants, especially for larger matrices.

Matrix Operations

Matrix operations, such as addition, multiplication, and finding inverses, are fundamental in linear algebra. These operations are essential for manipulating matrices to compute determinants. Familiarity with how these operations affect the determinant, such as how multiplying a row by a scalar affects the determinant value, is crucial for applying determinant theorems effectively.