A project manager is analyzing a resource allocation spreadsheet using a $3 imes 3$determinant to ensure budget balance. Given the cost matrix$C = \begin{vmatrix} 5 & 2 & 9 \ 3 & 1 & 4 \ 6 & 0 & 7 \end{vmatrix}$, 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 ${}3 imes 3$ 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 ${}3 imes 3$ force matrix $M = \begin{vmatrix} a_1 & b_1 & c_1 \\ a_2 & b_2 & c_2 \\ a_3 & b_3 & c_3 \end{vmatrix}$ 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 ${}3 imes 3$ 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 ${}3 \times 3$ 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.