교과서 질문
Write each linear system as a matrix equation in the form AX = B, where A is the coefficient matrix and B is the constant matrix.
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Ch. 6 - Matrices and Determinants
Blitzer8th EditionCollege AlgebraISBN: 9780136970514
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Table of contents
- Ch. P - Fundamental Concepts of Algebra311
- Ch. 1 - Equations and Inequalities398
- Ch. 2 - Functions and Graphs407
- Ch. 3 - Polynomial and Rational Functions306
- Ch. 4 - Exponential and Logarithmic Functions247
- Ch. 5 - Systems of Equations and Inequalities173
- Ch. 6 - Matrices and Determinants143
- Ch. 7 - Conic Sections124
- Ch. 8 - Sequences, Induction, and Probability251
7장, 문제 31
Solve each system of equations using matrices. Use Gaussian elimination with back-substitution or Gauss-Jordan elimination.
검증된 단계별 안내1
Write the system of equations as an augmented matrix. The system is:
\(\begin{cases} 3a - b - 4c = 3 \\ 2a - b + 2c = -8 \\ a + 2b - 3c = 9 \end{cases}\)
The augmented matrix is:
\(\left[ \begin{array}{ccc|c} 3 & -1 & -4 & 3 \\ 2 & -1 & 2 & -8 \\ 1 & 2 & -3 & 9 \end{array} \right]\)
Use Gaussian elimination to transform the matrix into an upper triangular form. Start by using the first row to eliminate the 'a' terms in the second and third rows. For example, multiply the third row by 3 and subtract the first row multiplied by 1 to eliminate the 'a' in the third row.
Continue the elimination process to get zeros below the leading coefficients in the first and second columns. This will give you a matrix in row echelon form, where the lower left part of the matrix has zeros.
Once the matrix is in upper triangular form, use back-substitution to solve for the variables starting from the last row. Solve for 'c' first, then substitute back to find 'b', and finally 'a'.
Alternatively, you can use Gauss-Jordan elimination to reduce the matrix to reduced row echelon form (RREF), where the matrix has leading 1s and zeros everywhere else in the variable columns, directly giving the solution for 'a', 'b', and 'c'.
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주요 개념
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Systems of Linear Equations
A system of linear equations consists of multiple linear equations involving the same set of variables. The goal is to find values for the variables that satisfy all equations simultaneously. Understanding how to represent and interpret these systems is fundamental before applying matrix methods.
추천 영상:
가이드 코스
4:27
Introduction to Systems of Linear Equations
Matrix Representation of Systems
Systems of linear equations can be expressed in matrix form as AX = B, where A is the coefficient matrix, X is the column matrix of variables, and B is the constants matrix. This representation simplifies the use of matrix operations to solve the system efficiently.
추천 영상:
가이드 코스
6:19Systems of Inequalities
Gaussian Elimination and Gauss-Jordan Elimination
Gaussian elimination transforms the augmented matrix into an upper triangular form to solve by back-substitution, while Gauss-Jordan elimination reduces it further to reduced row echelon form for direct solution. Both methods use row operations to systematically solve linear systems.
추천 영상:
가이드 코스
6:48Solving Systems of Equations - Elimination
관련 실천
교과서 질문
Write each linear system as a matrix equation in the form AX = B, where A is the coefficient matrix and B is the constant matrix.
675
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교과서 질문
In Exercises 31–36, use the alternative method for evaluating third-order determinants on here to evaluate each determinant.
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교과서 질문
In Exercises 27 - 36, find (if possible) the following matrices: a. AB b. BA 1 2 A = [1 2 3 4], B = 3 4
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교과서 질문
In Exercises 27 - 36, find (if possible) the following matrices: a. AB b. BA 1 - 1 4 1 1 0 A = 4 - 1 3 B = 1 2 4 2 0 - 2 1 - 1 3
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교과서 질문
In Exercises 31–36, use the alternative method for evaluating third-order determinants on here to evaluate each determinant.
808
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