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

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 \( \begin{bmatrix} a & b \\ c & d \end{bmatrix} \).
2. If the matrix is larger than 2x2, consider using cofactor expansion. Choose a row or column to expand along, ideally one with zeros to simplify calculations.
3. For cofactor expansion, calculate the determinant by multiplying each element of the chosen row or column by its cofactor, and sum these products. The cofactor is calculated as \((-1)^{i+j} \times \text{det}(M_{ij})\), where \(M_{ij}\) is the minor matrix obtained by removing the i-th row and j-th column.
4. If the matrix is triangular (upper or lower), the determinant is simply the product of the diagonal elements.
5. Apply any relevant determinant properties, such as the effect of row swaps (which change the sign of the determinant) or factoring out common factors from rows or columns, to simplify the calculation.

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. Determinants can be calculated using various methods, including cofactor expansion and row reduction.

Cofactor Expansion

Cofactor expansion is a method for calculating the determinant of a matrix by expressing it in terms of the determinants of 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.

Properties of Determinants

Determinants have several key properties that simplify their computation and understanding. 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 or column of zeros, its determinant is zero, indicating that the matrix is singular and not invertible.