In Exercises 31–36, use the alternative method for evaluating third-order determinants on here to evaluate each determinant. 1 5 61 4 51 9 10

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. For a 3x3 matrix, the determinant can be calculated using various methods, including cofactor expansion and the alternative method, which simplifies the process.

Cofactor expansion is a technique used to calculate the determinant of a matrix by breaking it down into smaller matrices. For a 3x3 matrix, the determinant can be computed by selecting any 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 highlights the recursive nature of determinants.

Third-order determinants refer specifically to the determinants of 3x3 matrices. The calculation involves a specific formula or method that accounts for the arrangement of elements in the matrix. Understanding how to evaluate third-order determinants is crucial in linear algebra, particularly in solving systems of equations and analyzing linear transformations.