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Expanding by Minors along the First Row to Evaluate a 3 × 3 Determinant
To evaluate a determinant by expanding by minors along the first row, each entry in the first row is multiplied by its corresponding minor (the determinant remaining after the entry's row and column are removed), and the results are combined using the alternating sign pattern . Specifically, for a determinant with first-row entries , , and , the value equals times the minor of , minus times the minor of , plus times the minor of .
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Intermediate Algebra @ OpenStax
Algebra
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Expanding by Minors along the First Row to Evaluate a 3 × 3 Determinant
As a logistics analyst verifying an algorithm that evaluates the determinant of a routing matrix by expanding by minors, you need to recall the alternating checkerboard sign pattern. Which of the following correctly represents this sign pattern for a determinant?
An analyst is auditing a financial risk model that involves evaluating the determinant of a matrix. According to the alternating sign pattern used for expansion by minors, the sign assigned to the entry in the top-left position (Row 1, Column 1) is negative.
Auditing Sign Patterns in Matrix Determinants
A logistics software developer is auditing a module that calculates the determinant of a shipping volume matrix. To verify the 'Expansion by Minors' algorithm, match each matrix position with its correct sign based on the alternating checkerboard pattern.
An office manager is verifying a spreadsheet macro that calculates the determinant of a matrix for a budget allocation model. The macro uses the method of expansion by minors, which requires applying an alternating checkerboard sign pattern. If the pattern begins with a positive sign in the top-left position (Row 1, Column 1), then the sign assigned to the entry in Row 1, Column 2 must be ____.
Learn After
A project manager is analyzing a resource allocation spreadsheet using a $3 imes 3C = \begin{vmatrix} 5 & 2 & 9 \ 3 & 1 & 4 \ 6 & 0 & 7 \end{vmatrix}$, which of the following expressions represents the correct signed term for the second entry in the first row when expanding by minors along that row?
An inventory manager is using a matrix to evaluate stock distribution across three warehouses. Arrange the following steps in the correct order to calculate the matrix's determinant by expanding by minors along the first row.
A structural engineer is analyzing a force matrix to determine the stability of a bridge support. To evaluate the determinant by expanding by minors along the first row, match each entry from the first row with the correct signed minor used in the expansion formula.
A data analyst is auditing a spreadsheet formula designed to calculate a determinant for a market trend model. When expanding the determinant along the first row, the analyst must ensure that the three resulting terms are combined using an alternating sign pattern of positive, _____, and positive.
A financial auditor is validating a custom software script that calculates the determinant of a matrix by expanding by minors along the first row. The script combines the three terms—each consisting of a first-row entry multiplied by its corresponding minor—by adding the first term, subtracting the second term, and subtracting the third term.