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

Verified step by step guidance

1. Identify the size of the matrix for which you need to find the determinant. If it's a 2x2 matrix, use the formula: \( \text{det}(A) = ad - bc \) for a matrix \( \begin{bmatrix} a & b \\ c & d \end{bmatrix} \).
2. For a 3x3 matrix, use the rule of Sarrus or the cofactor expansion method. The rule of Sarrus involves summing the products of the diagonals from the top left to the bottom right and subtracting the products of the diagonals from the top right to the bottom left.
3. If using cofactor expansion for a 3x3 matrix, choose a row or column to expand along. Calculate the determinant by multiplying each element by its cofactor and summing the results.
4. For larger matrices, reduce the matrix to a simpler form using row operations or expand using cofactors until you reach a 2x2 or 3x3 matrix.
5. Apply the determinant properties and theorems, such as the effect of row swaps, row multiplication, or row addition, to simplify the calculation if necessary.

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.