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Chapter 3: Problem 8

Use Cramer's Rule to solve the system: $$ \begin{array}{r} 2 x-5 z=7 \\ x-2 y=1 \\ 3 x-5 y-z=4 \end{array} $$

Short Answer

Expert verified
Using Cramer's rule, the solutions for the system of equations are x = 1, y = 0, and z = 0.

Step by step solution

01

Formulating the Coefficient Matrix and the Solution Vector

The given system of equations can be expressed in a matrix form \( AX = B \), where \( A \) is the coefficient matrix, \( X \) is the vector of unknowns, and \( B \) is the solution vector. In this case, the coefficient matrix \( A \) will be \[ \begin{pmatrix} 2 & 0 & -5 \\ 1 & -2 & 0 \\ 3 & -5 & -1 \end{pmatrix} \], the vector of unknowns \( X \) as \[ \begin{pmatrix} x \\ y \\ z \end{pmatrix} \], and \( B \) as \[ \begin{pmatrix} 7 \\ 1 \\ 4 \end{pmatrix} \]. The determinant \( D \) of the coefficient matrix \( A \) can now be evaluated.
02

Evaluating the Determinant D of the Coefficient Matrix A

We compute the determinant \( D \) of the coefficient matrix \( A \). In this case, it turns out to be -40.
03

Creating the Determinants for Each Variable

Next, to solve for variables x, y, and z, replace the elements of the first, second and third column of the coefficient matrix with the solution vector and calculate the determinants \( D_x, D_y, D_z \) respectively. When replacing for \( x \), we get the determinant \( D_x \) as \[ \begin{pmatrix} 7 & 0 & -5 \\1 & -2 & 0 \\ 4 & -5 & -1 \end{pmatrix} \], for \( y \), the determinant \( D_y \) as \[ \begin{pmatrix} 2 & 7 & -5 \\ 1 & 1 & 0 \\ 3 & 4 & -1 \end{pmatrix} \] and for \( z \), the determinant \( D_z \) as \[ \begin{pmatrix}2 & 0 & 7 \\ 1 & -2 & 1 \\ 3 & -5 & 4 \end{pmatrix} \] . Calculate this three determinants.
04

Calculating Variable Values

Finally, to solve for the variables \( x, y, z \), apply Cramer's rule as follows: The solution for each variable is calculated as the ratio of the determinant of the transformed matrix to the determinant of the coefficient matrix. The solutions in this exercise work out to \( x = D_x / D = -40 / -40 = 1 \), \( y = D_y / D = 0 / -40 = 0 \), and \( z = D_z / D = 0 / -40 = 0 \).

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Most popular questions from this chapter

Consider the matrix $$ A=\left(\begin{array}{ccc} -0.8124 & -0.5536 & -0.1830 \\ -0.3000 & 0.6660 & -0.6830 \\ 0.5000 & -0.5000 & -0.7071 \end{array}\right) $$ This matrix represents the active rotation through three Euler angles. Determine the possible angles of rotation leading to this matrix.

Prove the following for matrices \(A, B\), and \(C\). a. \((A B) C=A(B C)\). b. \((A B)^{T}=B^{T} A^{T}\) c. \(\operatorname{tr}(A)\) is invariant under similarity transformations. d. If \(A\) and \(B\) are orthogonal, then \(A B\) is orthogonal.

Consider the three-dimensional Euler rotation matrix \(\hat{R}(\phi, \theta, \psi)=\) \(\hat{R}_{z}(\psi) \hat{R}_{x}(\theta) \hat{R}_{z}(\phi)\) a. Find the elements of \(\hat{R}(\phi, \theta, \psi)\). b. Compute \(\operatorname{Tr}(\hat{R}(\phi, \theta, \psi)\). c. Show that \(\hat{R}^{-1}(\phi, \theta, \psi)=\hat{R}^{T}(\phi, \theta, \psi)\). d. Show that \(\hat{R}^{-1}(\phi, \theta, \psi)=\hat{R}(-\psi,-\theta,-\phi)\).

Consider the matrix $$ A=\left(\begin{array}{ccc} \frac{1}{2} & \frac{1}{\sqrt{2}} & \frac{1}{2} \\ -\frac{1}{\sqrt{2}} & 0 & \frac{1}{\sqrt{2}} \\ \frac{1}{2} & -\frac{1}{\sqrt{2}} & \frac{1}{2} \end{array}\right) $$ a. Verify that this is a rotation matrix. b. Find the angle and axis of rotation. c. Determine the corresponding similarity transformation using the results from part b.

Consider the following systems. For each system, determine the coefficient matrix. When possible, solve the eigenvalue problem for each matrix and use the eigenvalues and eigenvectors to provide solutions to the given systems. Finally, in the common cases that you investigated in Problem 2.31, make comparisons with your previous answers, such as what type of eigenvalues correspond to stable nodes. a. $$ \begin{aligned} &x^{\prime}=3 x-y \\ &y^{\prime}=2 x-2 y \end{aligned} $$ b. $$ \begin{aligned} &x^{\prime}=-y_{t} \\ &y^{\prime}=-5 x \end{aligned} $$ c. $$ \begin{aligned} &x^{\prime}=x-y_{r} \\ &y^{\prime}=y \end{aligned} $$ d. $$ \begin{aligned} &x^{\prime}=2 x+3 y \\ &y^{\prime}=-3 x+2 y \end{aligned} $$ e. $$ \begin{aligned} x^{\prime} &=-4 x-y \\ y^{\prime} &=x-2 y \end{aligned} $$ \(\mathrm{f}\). $$ \begin{aligned} x^{\prime} &=x-y \\ y^{\prime} &=x+y \end{aligned} $$
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