Evaluate each determinant in Exercises 1–10. - 7 142 - 4

The determinant of a 2x2 matrix is calculated using the formula |A| = ad - bc, where A is the matrix [[a, b], [c, d]]. This value provides important information about the matrix, such as whether it is invertible (a non-zero determinant indicates invertibility) and the area of the parallelogram formed by its column vectors.

A matrix is a rectangular array of numbers arranged in rows and columns. In this case, the matrix is represented as [[-7, 14], [2, -4]]. Understanding how to read and interpret matrices is crucial for performing operations like finding determinants, as well as for applications in systems of equations and transformations.

Determinants have several key properties, including linearity, the effect of row operations, and the relationship to matrix multiplication. For instance, swapping two rows of a matrix changes the sign of the determinant, while multiplying a row by a scalar multiplies the determinant by that scalar. These properties are essential for simplifying calculations and understanding the behavior of matrices.