In Exercises 31–36, use the alternative method for evaluating third-order determinants on here to evaluate each determinant. 0.5 7 50.5 3 90.5 1 3

Verified step by step guidance

1. Identify the matrix as a 3x3 matrix: \( \begin{bmatrix} 0.5 & 7 & 5 \\ 0.5 & 3 & 9 \\ 0.5 & 1 & 3 \end{bmatrix} \).
2. Use the alternative method for evaluating third-order determinants, which involves expanding along a row or column.
3. Choose the first column for expansion: \( 0.5 \times \begin{vmatrix} 3 & 9 \\ 1 & 3 \end{vmatrix} - 0.5 \times \begin{vmatrix} 7 & 5 \\ 1 & 3 \end{vmatrix} + 0.5 \times \begin{vmatrix} 7 & 5 \\ 3 & 9 \end{vmatrix} \).
4. Calculate each 2x2 determinant: \( \begin{vmatrix} 3 & 9 \\ 1 & 3 \end{vmatrix} = 3 \times 3 - 9 \times 1 \), \( \begin{vmatrix} 7 & 5 \\ 1 & 3 \end{vmatrix} = 7 \times 3 - 5 \times 1 \), \( \begin{vmatrix} 7 & 5 \\ 3 & 9 \end{vmatrix} = 7 \times 9 - 5 \times 3 \).
5. Substitute the values of the 2x2 determinants back into the expression and simplify to find the determinant of the 3x3 matrix.

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

Third-order Determinants

Third-order determinants refer specifically to determinants of 3x3 matrices. The calculation involves a specific formula that incorporates the elements of the matrix in a structured way, often using the method of minors and cofactors. Understanding how to manipulate these matrices and apply the determinant formula is crucial for solving systems of equations and analyzing linear transformations.

Cofactor Expansion

Cofactor expansion is a technique used to calculate the determinant of a matrix by breaking it down into smaller matrices. For a 3x3 matrix, this 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 and provides a systematic approach to finding determinants.