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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0:59 minutes

Problem 1

Textbook Question

Evaluate each determinant in Exercises 1–10. 5 72 3

Verified step by step guidance

Identify the matrix as a 2x2 matrix with elements a=5, b=7, c=2, d=3.

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

Substitute the values into the formula: det(A) = (5)(3) - (7)(2).

Calculate the products: (5)(3) and (7)(2).

Subtract the second product from the first to find the determinant.

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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 |A| = ad - bc, where A = |a b| and |c d|. For the matrix given, the determinant is computed as (5)(3) - (7)(2), which simplifies to 15 - 14 = 1. The determinant provides important information about the matrix, including whether it is invertible.

Matrix Representation

A matrix is a rectangular array of numbers arranged in rows and columns. In this case, the matrix is represented as two rows and two columns, with the first row containing the numbers 5 and 7, and the second row containing 2 and 3. Understanding how to read and interpret matrices is essential for performing operations such as finding determinants.

Properties of Determinants

Determinants have several key properties, including that a matrix is invertible if and only if its determinant is non-zero. Additionally, the determinant changes sign if two rows are swapped, and if a row is multiplied by a scalar, the determinant is also multiplied by that scalar. These properties are crucial for understanding the implications of the determinant in linear algebra.