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Ch. 6 - Matrices and Determinants
Blitzer - College Algebra 8th Edition
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Tutti i libri di testoBlitzer 8th EditionCh. 6 - Matrices and DeterminantsProblema 41
Capitolo 7, Problema 41

In Exercises 37–44, use Cramer's Rule to solve each system. {x+y+z=4x−2y+z=7x+3y+2z=4\(\begin{cases}\) x + y + z = 4 \\ x - 2y + z = 7 \\ x + 3y + 2z = 4 \(\end{cases}\)

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Write the system of equations in matrix form as \(A\mathbf{x} = \mathbf{b}\), where \(A\) is the coefficient matrix, \(\mathbf{x}\) is the column vector of variables, and \(\mathbf{b}\) is the constants vector. For this system, \(A = \begin{bmatrix} 1 & 1 & 1 \\ 1 & -2 & 1 \\ 1 & 3 & 2 \end{bmatrix}\), \(\mathbf{x} = \begin{bmatrix} x \\ y \\ z \end{bmatrix}\), and \(\mathbf{b} = \begin{bmatrix} 4 \\ 7 \\ 4 \end{bmatrix}\).
Calculate the determinant of the coefficient matrix \(A\), denoted as \(\det(A)\). This is essential because Cramer's Rule requires \(\det(A) \neq 0\) to have a unique solution.
Form three new matrices \(A_x\), \(A_y\), and \(A_z\) by replacing the respective columns of \(A\) with the constants vector \(\mathbf{b}\). Specifically, \(A_x\) replaces the first column, \(A_y\) replaces the second column, and \(A_z\) replaces the third column with \(\mathbf{b}\).
Calculate the determinants of these new matrices: \(\det(A_x)\), \(\det(A_y)\), and \(\det(A_z)\).
Use Cramer's Rule to find each variable: \(x = \frac{\det(A_x)}{\det(A)}\), \(y = \frac{\det(A_y)}{\det(A)}\), and \(z = \frac{\det(A_z)}{\det(A)}\). These fractions give the solution to the system.

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Cramer's Rule

Cramer's Rule is a method for solving systems of linear equations using determinants. It applies when the system has the same number of equations as unknowns and the coefficient matrix has a non-zero determinant. Each variable is found by replacing the corresponding column in the coefficient matrix with the constants vector and calculating the determinant ratio.
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Cramer's Rule - 2 Equations with 2 Unknowns

Determinants of Matrices

The determinant is a scalar value that can be computed from a square matrix and provides important properties about the matrix, such as invertibility. For a 3x3 matrix, the determinant is calculated using a specific formula involving the elements of the matrix. A non-zero determinant indicates the system has a unique solution.
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Determinants of 2×2 Matrices

Systems of Linear Equations

A system of linear equations consists of multiple linear equations with the same variables. Solving the system means finding values for the variables that satisfy all equations simultaneously. Methods include substitution, elimination, matrix operations, and Cramer's Rule, especially useful for small systems.
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Introduction to Systems of Linear Equations
Pratica correlata
Domanda del libro di testo

Perform the indicated matrix operations given that A, B and C are defined as follows. If an operation is not defined, state the reason.

A=[40−3501],B=[51−2−2],C=[1−1−11]A=\(\begin{bmatrix}\)4 & 0\\ -3 & 5\\ 0 & 1\(\end{bmatrix}\),B=\(\begin{bmatrix}\)5 & 1\\ -2 & -2\(\end{bmatrix}\),C=\(\begin{bmatrix}\)1 & -1\\ -1 & 1\(\end{bmatrix}\)

A - C

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Domanda del libro di testo

In Exercises 37 - 44, perform the indicated matrix operations given that A, B and C are defined as follows. If an operation is not defined, state the reason.

A=[40−3501],B=[51−2−2],C=[1−1−11]A=\(\begin{bmatrix}\)4 & 0\\ -3 & 5\\ 0 & 1\(\end{bmatrix}\),B=\(\begin{bmatrix}\)5 & 1\\ -2 & -2\(\end{bmatrix}\),C=\(\begin{bmatrix}\)1 & -1\\ -1 & 1\(\end{bmatrix}\)

A(BC)

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Domanda del libro di testo

Solve the system: (Hint: Let A = ln w, B = ln x, C = ln y, and D = ln z. Solve the system for A, B, C, and D. Then use the logarithmic equations to find w, x, y, and z.)

{2lnw+lnx+3lny−2lnz=−64lnw+3lnx+lny−lnz=−2lnw+lnx+lny+lnz=−5lnw+lnx−lny−lnz=5\(\begin{cases}\) 2 \(\ln\) w + \(\ln\) x + 3 \(\ln\) y - 2 \(\ln\) z = -6 \\ 4 \(\ln\) w + 3 \(\ln\) x + \(\ln\) y - \(\ln\) z = -2 \\ \(\ln\) w + \(\ln\) x + \(\ln\) y + \(\ln\) z = -5 \\ \(\ln\) w + \(\ln\) x - \(\ln\) y - \(\ln\) z = 5 \(\end{cases}\)

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Domanda del libro di testo

In Exercises 37–44, use Cramer's Rule to solve each system. {x+2z=42y−z=52x+3y=13\(\begin{cases}\) x + 2z = 4 \\ 2y - z = 5 \\ 2x + 3y = 13 \(\end{cases}\)

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Domanda del libro di testo

Find the cubic function f(x) = ax³ + bx² + cx + d for which ƒ( − 1) = 0, ƒ(1) = 2, ƒ(2) = 3, and ƒ(3) = 12.

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Domanda del libro di testo

a. Write each linear system as a matrix equation in the form AX = B. b. Solve the system using the inverse that is given for the coefficient matrix.

{w−x+2y=−3x−y+z=4−w+x−y+2z=2−x+y−2z=−4The inverse of [1−12001−11−11−120−11−2] is [00−1−1141312120−10−1]\(\begin{cases}\) w - x + 2y \(\quad\)\(\quad\) = -3 \\ \(\quad\[\quad\) x - y + z = 4 \\ -w + x - y + 2z = 2 \\ \(\quad\]\quad\) -x + y - 2z = -4 \(\end{cases}\) \\ \(\text{The inverse of }\) \(\begin{bmatrix}\) 1 & -1 & 2 & 0 \\ 0 & 1 & -1 & 1 \\ -1 & 1 & -1 & 2 \\ 0 & -1 & 1 & -2 \(\end{bmatrix}\) \(\text{ is }\) \(\begin{bmatrix}\) 0 & 0 & -1 & -1 \\ 1 & 4 & 1 & 3 \\ 1 & 2 & 1 & 2 \\ 0 & -1 & 0 & -1 \(\end{bmatrix}\)

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