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

Verified step by step guidance

1. Identify the 2x2 matrix: \( \begin{bmatrix} -7 & 14 \\ 2 & -4 \end{bmatrix} \).
2. Recall the formula for the determinant of a 2x2 matrix: \( ad - bc \), where \( a, b, c, \) and \( d \) are the elements of the matrix.
3. Assign the values: \( a = -7 \), \( b = 14 \), \( c = 2 \), and \( d = -4 \).
4. Substitute these values into the determinant formula: \( (-7)(-4) - (14)(2) \).
5. Simplify the expression by performing the multiplications and then the subtraction.

Key Concepts

Determinant of a 2x2 Matrix

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.

Matrix Representation

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.

Properties of Determinants

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.