Table of contents
- 0. Review of Algebra4h 16m
- 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 19m
- 10. Combinatorics & Probability1h 45m
7. Systems of Equations & Matrices
Determinants and Cramer's Rule
1:38 minutes
Problem 53c
Textbook Question
Textbook QuestionUse the determinant theorems to evaluate each determinant. See Example 4.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Determinants
A determinant is a scalar value that can be computed from the elements of a square matrix. It provides important information about the matrix, such as whether it is invertible and the volume scaling factor of the linear transformation represented by the matrix. Determinants can be calculated using various methods, including cofactor expansion and row reduction.
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Cofactor Expansion
Cofactor expansion is a method for calculating the determinant of a matrix by expressing it in terms of the determinants of smaller matrices. This involves selecting a row or column, multiplying each element by its corresponding cofactor (which is the determinant of the submatrix formed by removing the row and column of that element, adjusted by a sign), and summing these products. This technique is particularly useful for larger matrices.
Properties of Determinants
Determinants have several key properties that simplify their computation and understanding. For instance, the determinant of a product of matrices equals the product of their determinants, and swapping two rows of a matrix changes the sign of the determinant. Additionally, if a matrix has a row or column of zeros, its determinant is zero, indicating that the matrix is singular and not invertible.
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