Evaluate each determinant in Exercises 1–10. - 5 - 1- 2 - 7

Verified step by step guidance

1. Identify the elements of the 2x2 matrix: a = -5, b = -1, c = -2, d = -7.
2. Recall the formula for the determinant of a 2x2 matrix: det(A) = ad - bc.
3. Substitute the values into the formula: det(A) = (-5)(-7) - (-1)(-2).
4. Calculate the product of a and d: (-5) * (-7).
5. Calculate the product of b and c: (-1) * (-2).

Key Concepts

Determinant of a 2x2 Matrix

The determinant of a 2x2 matrix is calculated using the formula ad - bc, where the matrix is represented as [[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 the context of determinants, a 2x2 matrix consists of four elements, which can be denoted as [[a, b], [c, d]]. Understanding how to represent and manipulate matrices is crucial for performing operations like finding determinants and solving systems of equations.

Properties of Determinants

Determinants have several key properties, including linearity, the effect of row operations, and the relationship to matrix invertibility. For example, 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.