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Chapter 9: Problem 6

For the set of equations $$\begin{array}{l}2 x_{2}+5 x_{3}=1 \\\2 x_{1}+x_{2}+x_{3}=1 \\\3 x_{1}+x_{2}=2 \end{array}$$ (a) Compute the determinant. (b) Use Cramer's rule to solve for the \(X\) 's. (c) Substitute your results back into the original equation to check your results.

Short Answer

Expert verified
The determinant of matrix A is \(\text{det}(A) = -1\). Using Cramer's rule, we find the values of \(x_1\), \(x_2\), and \(x_3\) to be: \[x_1 = \frac{\text{det}(A_1)}{\text{det}(A)} = \frac{1}{-1} = -1\] \[x_2 = \frac{\text{det}(A_2)}{\text{det}(A)} = \frac{-1}{-1} = 1\] \[x_3 = \frac{\text{det}(A_3)}{\text{det}(A)} = \frac{-4}{-1} = 4\] Substituting the values of \(x_1\), \(x_2\), and \(x_3\) back into the original system of equations, we verify that the solution is correct: \(2(1)+5(4) = 1\) \(2(-1)+1+4 = 1\) \(3(-1)+1 = 2\) Thus, the solution to the system of equations is \(x_1 = -1\), \(x_2 = 1\), and \(x_3 = 4\).

Step by step solution

01

Write down the coefficient matrix and compute the determinant

First, let's write down the coefficient matrix A, for the given system of equations: \[A = \begin{pmatrix} 0 & 2 & 5 \\\ 2 & 1 & 1 \\\ 3 & 1 & 0 \end{pmatrix}\] Now compute the determinant of matrix A: \[\text{det}(A) = \left|\begin{array}{ccc} 0 & 2 & 5 \\\ 2 & 1 & 1 \\\ 3 & 1 & 0 \end{array}\right|\]
02

Compute \(\text{det}(A_1)\), \(\text{det}(A_2)\), and \(\text{det}(A_3)\) by Cramer's rule

Cramer's rule states that we can find the solution for \(x_1\), \(x_2\), and \(x_3\) by computing the determinants of the matrices \(A_1\), \(A_2\), and \(A_3\): \(A_1\) is obtained by replacing the first column of A with the constant terms of the system of equations: \[A_1 = \begin{pmatrix} 1 & 2 & 5 \\\ 1 & 1 & 1 \\\ 2 & 1 & 0 \end{pmatrix}\] \(A_2\) is obtained by replacing the second column of A with the constant terms of the system of equations: \[A_2 = \begin{pmatrix} 0 & 1 & 5 \\\ 2 & 1 & 1 \\\ 3 & 2 & 0 \end{pmatrix}\] \(A_3\) is obtained by replacing the third column of A with the constant terms of the system of equations: \[A_3 = \begin{pmatrix} 0 & 2 & 1 \\\ 2 & 1 & 1 \\\ 3 & 1 & 2 \end{pmatrix}\] Now compute the determinants of the matrices \(A_1\), \(A_2\), and \(A_3\): \(\text{det}(A_1) = \left|\begin{array}{ccc} 1 & 2 & 5 \\\ 1 & 1 & 1 \\\ 2 & 1 & 0 \end{array}\right|\) \(\text{det}(A_2) = \left|\begin{array}{ccc} 0 & 1 & 5 \\\ 2 & 1 & 1 \\\ 3 & 2 & 0 \end{array}\right|\) \(\text{det}(A_3) = \left|\begin{array}{ccc} 0 & 2 & 1 \\\ 2 & 1 & 1 \\\ 3 & 1 & 2 \end{array}\right|\)
03

Calculate the values of \(x_1\), \(x_2\), and \(x_3\) using Cramer's rule

Using Cramer's rule, we can find the values of \(x_1\), \(x_2\), and \(x_3\) as follows: \[x_1 = \frac{\text{det}(A_1)}{\text{det}(A)}\] \[x_2 = \frac{\text{det}(A_2)}{\text{det}(A)}\] \[x_3 = \frac{\text{det}(A_3)}{\text{det}(A)}\] Plug in the values of \(\text{det}(A_1)\), \(\text{det}(A_2)\), \(\text{det}(A_3)\), and \(\text{det}(A)\) into the formulas for x1, x2, and x3 to find their values.
04

Substitute the obtained values of \(x_1\), \(x_2\), and \(x_3\) back into the original system of equations

Now that we have the values of \(x_1\), \(x_2\), and \(x_3\), we can substitute them back into the original system of equations to verify if the solution is correct: \[2 x_{2}+5 x_{3}=1\] \[2 x_{1}+x_{2}+x_{3}=1\] \[3 x_{1}+x_{2}=2\] Make sure that all the three equations are satisfied after substitution. If they are, then the solution is correct.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Determinants
A determinant is a special scalar value that can be computed from a square matrix. It provides important information about the matrix, such as whether the matrix is invertible or singular. In the case of linear equations, the determinant can help determine if a unique solution exists.

To calculate the determinant of a 3x3 matrix like matrix A from our exercise, you apply a method that includes multiplying and subtracting products of the matrix's elements. For matrix A, the determinant would be computed as follows:
det(A) = a_{11}(a_{22}a_{33} - a_{23}a_{32}) - a_{12}(a_{21}a_{33} - a_{23}a_{31}) + a_{13}(a_{21}a_{32} - a_{22}a_{31})
Where, for example, a_{11} is the element in the first row and first column of matrix A. The determinant tells us if the system of equations has a unique solution (non-zero determinant) or not (zero determinant).
System of Linear Equations
A system of linear equations consists of multiple linear equations that share a common set of variables. These systems can be solved using various methods, one of which is Cramer's rule, which applies when you have the same number of independent equations as variables and the determinant of the coefficient matrix is non-zero. Cramer's rule is particularly helpful for solving small systems of equations with a unique solution.

In our exercise, the system consists of three equations with three variables represented as x1, x2, and x3. To apply Cramer's rule effectively, we ensure that the system is consistent and has a unique solution by verifying that the determinant of the coefficient matrix is not equal to zero. As long as this condition holds, Cramer's rule can be used to find each variable by substituting the columns of the coefficient matrix A with the constants from the equations and evaluating the determinant of each resulting matrix.
Matrix Algebra
Matrix algebra is a branch of mathematics that deals with matrices and their operations, including addition, subtraction, multiplication, and the calculation of determinants. It is an invaluable tool for solving systems of linear equations, as it can simplify many computations and offer methods like Cramer's rule for finding solutions efficiently.

In our exercise, we manipulate the coefficient matrix A to create new matrices A1, A2, and A3, which are essential for applying Cramer's rule. The matrices A1, A2, and A3 are formed by replacing each column of A with the vector of constants, sequentially, from the system of equations. The application of matrix algebra simplifies the process of solving for x1, x2, and x3, illustrating the usefulness of matrices in handling linear algebra problems.

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Most popular questions from this chapter

Given the system of equations \\[\begin{aligned}0.77 x_{1}+x_{2} &=14.25 \\\1.2 x_{1}+1.7 x_{2} &=20\end{aligned}\\] (a) Solve graphically and check your results by substituting them back into the equations. (b) On the basis of the graphical solution, what do you expect regarding the condition of the system? (c) Compute the determinant. (d) Solve by the elimination of unknowns.

Given the equations \\[\begin{array}{l}10 x_{1}+2 x_{2}-x_{3}=27 \\\\-3 x_{1}-6 x_{2}+2 x_{3}=-61.5 \\ x_{1}+x_{2}+5 x_{3}=-21.5\end{array}\\] (a) Solve by naive Gauss elimination. Show all steps of the computation. (b) Substitute your results into the original equations to check your answers.

Use the graphical method to solve \\[ \begin{array}{l} 2 x_{1}-6 x_{2}=-18 \\ -x_{1}+8 x_{2}=40 \end{array} \\] Check your results by substituting them back into the equations.

A number of matrices are defined as \([A]=\left[\begin{array}{ll}4 & 5 \\ 1 & 2 \\ 5 & 6\end{array}\right] \quad[B]=\left[\begin{array}{lll}4 & 3 & 7 \\ 1 & 2 & 6 \\ 2 & 0 & 4\end{array}\right]\) \(\\{C\\}=\left\\{\begin{array}{l}3 \\ 5 \\ 1\end{array}\right\\} \quad[D]=\left[\begin{array}{cccc}9 & 4 & 3 & -6 \\ 2 & -1 & 6 & 5\end{array}\right]\) \([E]=\left[\begin{array}{lll}1 & 5 & 9 \\ 7 & 2 & 3 \\ 4 & 0 & 6\end{array}\right]\) \([F]=\left[\begin{array}{lll}2 & 0 & 1 \\ 1 & 7 & 3\end{array}\right] \quad\lfloor G\rfloor=\lfloor 7 \quad 5 \quad 4\rfloor\) Answer the following questions regarding these matrices: (a) What are the dimensions of the matrices? (b) Identify the square, column, and row matrices. (c) What are the values of the elements: \(a_{12}, b_{23}, d_{32}, e_{22}, f_{12}, g_{12} ?\) (d) Perform the following operations: (1) \([E]+[B]\) (3) \([B]-[E]\) (2) \([A]+[F]\) \((4) 7 \times[B]\) (5) \([A] \times[B]\) (9) \([A] \times\\{C\\}\) (6)\(\\{C\\}^{T}\) \((10)[I] \times[B]\) (7) \([B] \times[A]\) (11)\([E]^{T}[E]\) (8) \([D]^{T}\) (12)\(\\{C\\}^{T}\\{C\\}\)

(a) Write the following set of equations in matrix form: \(40=5 x_{3}+2 x_{1}\) \\[ \begin{array}{l} 10-x_{2}=x_{3} \\ 3 x_{2}+8 x_{1}=20 \end{array} \\] (b) Write the transpose of the matrix of coefficients.
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