Minor of an Entry in a $3 \times 3$ Determinant

In a professional data analysis scenario, a researcher uses a ${}2 imes 2$ matrix $\begin{bmatrix} a & b \\ c & d \end{bmatrix}$ to model a system's efficiency. Which of the following formulas correctly represents the determinant of this matrix?

A quality control supervisor uses the ${}2 imes 2$ matrix $\begin{bmatrix} a & b \\ c & d \end{bmatrix}$ to monitor equipment performance across two shifts. Match the following mathematical expressions with their correct role in calculating the matrix's determinant.

An inventory manager uses a ${}2 \times 2$ matrix $\begin{bmatrix} a & b \\ c & d \end{bmatrix}$ to track shipping costs between two regions. To determine the model's stability, the manager must calculate the determinant of the matrix. Complete the standard formula used for this calculation: $\begin{vmatrix} a & b \\ c & d \end{vmatrix} =$ ____.

A data analyst is documenting the standard procedure for calculating the determinant of a ${}2 imes 2$ matrix $\begin{bmatrix} a & b \\ c & d \end{bmatrix}$ used in a logistics model. Arrange the following steps in the correct order to describe this calculation process.

A logistics coordinator is verifying a formula in a spreadsheet designed to evaluate supply chain efficiency using a ${}2 \times 2$ matrix $\begin{bmatrix} a & b \\ c & d \end{bmatrix}$. The correct formula to calculate the determinant of this matrix is $\begin{vmatrix} a & b \\ c & d \end{vmatrix} = bc - ad$.