Evaluate each determinant.

Verified step by step guidance

1. Identify the size of the matrix for which you need to evaluate the determinant. For a 2x2 matrix, use the formula: \( \text{det}(A) = ad - bc \) where \( A = \begin{bmatrix} a & b \\ c & d \end{bmatrix} \).
2. For a 3x3 matrix, use the rule of Sarrus or the cofactor expansion method. The rule of Sarrus is applicable only for 3x3 matrices and involves summing the products of the diagonals from left to right and subtracting the products of the diagonals from right to 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 (4x4 or more), use the method of cofactor expansion recursively or apply row reduction to simplify the matrix to an upper triangular form, then multiply the diagonal elements.
5. Remember that properties of determinants can simplify calculations: swapping two rows changes the sign of the determinant, multiplying a row by a scalar multiplies the determinant by that scalar, and adding a multiple of one row to another does not change the determinant.

Key Concepts

Determinant of a Matrix

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). The determinant can also be interpreted geometrically as the volume scaling factor of the linear transformation described 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 a matrix has a row of zeros, its determinant is zero, which indicates that the matrix is singular.

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. This method is particularly useful for larger matrices.