In Exercises 31–36, use the alternative method for evaluating third-order determinants on here to evaluate each determinant. $\(\begin{vmatrix}\)0.5 & 7 & 5 \\0.5 & 3 & 9 \\0.5 & 1 & 3\(\end{vmatrix}\)$

Verified step by step guidance

1. Step 1: Write down the given 3x3 determinant: \[\left| \begin{array}{ccc} 0.5 & 7 & 5 \\ 0.5 & 3 & 9 \\ 0.5 & 1 & 3 \end{array} \right|\]
2. Step 2: Use the alternative method for evaluating third-order determinants, which involves expanding along the first row or using the rule of Sarrus. Here, we will use the rule of Sarrus: - Repeat the first two columns to the right of the matrix: \[\begin{array}{ccc|cc} 0.5 & 7 & 5 & 0.5 & 7 \\ 0.5 & 3 & 9 & 0.5 & 3 \\ 0.5 & 1 & 3 & 0.5 & 1 \end{array}\]
3. Step 3: Calculate the sum of the products of the diagonals from top-left to bottom-right: \[ (0.5 \times 3 \times 3) + (7 \times 9 \times 0.5) + (5 \times 0.5 \times 1) \]
4. Step 4: Calculate the sum of the products of the diagonals from bottom-left to top-right: \[ (0.5 \times 3 \times 5) + (1 \times 9 \times 0.5) + (3 \times 0.5 \times 7) \]
5. Step 5: Subtract the sum from Step 4 from the sum in Step 3 to find the value of the determinant: \[ \text{Determinant} = \left[ (0.5 \times 3 \times 3) + (7 \times 9 \times 0.5) + (5 \times 0.5 \times 1) \right] - \left[ (0.5 \times 3 \times 5) + (1 \times 9 \times 0.5) + (3 \times 0.5 \times 7) \right] \]

Key Concepts

Third-Order Determinants

A third-order determinant is a scalar value calculated from a 3x3 matrix. It helps determine properties like matrix invertibility and solutions to systems of equations. The determinant is computed using specific formulas or methods, such as expansion by minors or the alternative method.

Alternative Method for Evaluating Determinants

The alternative method, often called the Rule of Sarrus, is a shortcut for finding the determinant of a 3x3 matrix. It involves summing the products of diagonals from left to right and subtracting the products of diagonals from right to left, simplifying the calculation process.

Matrix Representation of Systems of Equations

Matrices can represent systems of linear equations compactly, where each row corresponds to an equation and each column to a variable or constant. Evaluating the determinant of the coefficient matrix helps determine if the system has a unique solution, infinite solutions, or none.