Evaluate each determinant.

Verified step by step guidance

1. Identify the size of the matrix for which you need to evaluate the determinant. For example, a 2x2 or 3x3 matrix.
2. For a 2x2 matrix, use the formula: \( \text{det}(A) = ad - bc \) where the matrix is \( \begin{bmatrix} a & b \\ c & d \end{bmatrix} \).
3. For a 3x3 matrix, use the rule of Sarrus or the cofactor expansion method. For the rule of Sarrus, extend the first two columns of the matrix to the right and calculate the sum of the products of the diagonals from top left to bottom right, then subtract the sum of the products of the diagonals from bottom left to top right.
4. For the cofactor expansion method on a 3x3 matrix, choose a row or column to expand along. Calculate the determinant by summing the products of each element in the row or column with its corresponding cofactor.
5. Simplify the expression obtained from the determinant formula to find the determinant value.

Key Concepts

Determinants in Linear Algebra

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 also be used to calculate the area or volume of geometric shapes defined by the matrix.

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 (or columns) of a matrix are identical, the determinant is zero.

Cofactor Expansion

Cofactor expansion is a method for calculating 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 cofactor (which is the determinant of the submatrix formed by removing the element's row and column), and summing these products. It is particularly useful for larger matrices.