Table of contents

0. Review of Algebra4h 18m
1. Equations & Inequalities3h 18m
2. Graphs of Equations43m
3. Functions2h 17m
4. Polynomial Functions1h 44m
5. Rational Functions1h 23m
6. Exponential & Logarithmic Functions2h 28m
7. Systems of Equations & Matrices4h 6m
8. Conic Sections2h 23m
9. Sequences, Series, & Induction1h 22m
10. Combinatorics & Probability1h 45m

7. Systems of Equations & Matrices

Determinants and Cramer's Rule

College Algebra

7. Systems of Equations & Matrices

Determinants and Cramer's Rule

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1:12 minutes

Problem 7

Textbook Question

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

Verified step by step guidance

Identify the elements of the 2x2 matrix: a = -5, b = -1, c = -2, d = -7.

Recall the formula for the determinant of a 2x2 matrix: det(A) = ad - bc.

Substitute the values into the formula: det(A) = (-5)(-7) - (-1)(-2).

Calculate the product of a and d: (-5) * (-7).

Calculate the product of b and c: (-1) * (-2).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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.