Domanda del libro di testo
In Exercises 14–27, perform the indicated matrix operations given that and D are defined as follows. If an operation is not defined, state the reason. -5(A+D)
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Ch. 6 - Matrices and Determinants
Blitzer8th EditionCollege AlgebraISBN: 9780136970514
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TOC.tableOfContents
- 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
Capitolo 7, Problema 19
For Exercises 11–22, use Cramer's Rule to solve each system.
Guida verificata passo dopo passo1
Write the system of equations in standard form:
\(\begin{cases} 3x - 4y = 4 \\ 2x + 2y = 12 \end{cases}\)
Identify the coefficients for the variables and constants:
\(A = \begin{bmatrix} 3 & -4 \\ 2 & 2 \end{bmatrix}\),
\(\mathbf{x} = \begin{bmatrix} x \\ y \end{bmatrix}\),
\(\mathbf{b} = \begin{bmatrix} 4 \\ 12 \end{bmatrix}\)
Calculate the determinant of matrix \(A\), denoted as \(D\):
\(D = \det(A) = (3)(2) - (-4)(2)\)
Form matrices \(A_x\) and \(A_y\) by replacing the respective columns of \(A\) with the constants vector \(\mathbf{b}\):
\(A_x = \begin{bmatrix} 4 & -4 \\ 12 & 2 \end{bmatrix}\),
\(A_y = \begin{bmatrix} 3 & 4 \\ 2 & 12 \end{bmatrix}\)
Calculate the determinants \(D_x = \det(A_x)\) and \(D_y = \det(A_y)\), then use Cramer's Rule to find the solutions:
\(x = \frac{D_x}{D}\) and \(y = \frac{D_y}{D}\)
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Cramer's Rule
Cramer's Rule is a method for solving systems of linear equations using determinants. It applies to systems with the same number of equations and unknowns, where the solution for each variable is found by replacing the corresponding column of the coefficient matrix with the constants vector and dividing by the determinant of the coefficient matrix.
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Cramer's Rule - 2 Equations with 2 Unknowns
Determinants of 2x2 Matrices
The determinant of a 2x2 matrix [[a, b], [c, d]] is calculated as ad - bc. This value is crucial in Cramer's Rule, as it determines whether the system has a unique solution (non-zero determinant) or not. Calculating determinants accurately is essential for applying Cramer's Rule.
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4:36Determinants of 2×2 Matrices
Systems of Linear Equations
A system of linear equations consists of two or more linear equations with the same variables. Understanding how to represent and manipulate these systems, such as writing them in matrix form, is fundamental for applying methods like Cramer's Rule to find solutions.
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4:27Introduction to Systems of Linear Equations
Pratica correlata
Domanda del libro di testo
For Exercises 11–22, use Cramer's Rule to solve each system.
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Domanda del libro di testo
In Exercises 1 - 24, use Gaussian Elimination to find the complete solution to each system of equations, or show that none exists.
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Domanda del libro di testo
In Exercises 14–27, perform the indicated matrix operations given that and D are defined as follows. If an operation is not defined, state the reason. 3A+2D
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Domanda del libro di testo
Let and . Solve each matrix equation for X. 2X + A = B
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Domanda del libro di testo
In Exercises 19–20, a few steps in the process of simplifying the given matrix to row-echelon form, with 1s down the diagonal from upper left to lower right, and 0s below the 1s, are shown. Fill in the missing numbers in the steps that are shown.
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