Evaluate each determinant. See Example 3.

Verified step by step guidance

1. Identify the size of the matrix. If it's a 2x2 matrix, use the formula for the determinant: \( \text{det}(A) = ad - bc \) for a matrix \( A = \begin{bmatrix} a & b \\ c & d \end{bmatrix} \).
2. If the matrix is larger than 2x2, use cofactor expansion to evaluate the determinant. Choose a row or column to expand along, typically the one with the most zeros to simplify calculations.
3. For each element in the chosen row or column, calculate the minor by removing the row and column of that element, then find the determinant of the resulting smaller matrix.
4. Multiply each minor by its corresponding cofactor, which is \((-1)^{i+j}\) times the minor, where \(i\) and \(j\) are the row and column indices of the element.
5. Sum all the products of elements and their cofactors to find the determinant of the original matrix.

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 or row reduction.

Matrix Operations

Matrix operations, including addition, subtraction, and multiplication, are fundamental in linear algebra. Understanding how to manipulate matrices is essential for evaluating determinants, as the properties of matrices directly affect their determinants. For example, the determinant of a product of matrices equals the product of 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.