Math Problem Statement
Write the given system of equations as a matrix equation and solve using inverses. Complete parts a through c. x 1x1+4 x 2=k1 x 2+x3=k2 3 x 1−x2−14x3=k3. What are x 1,x2,x3 when k 1=32,k 2−3,and k 3=142?
Solution
The system of equations given is:
x_1 + 4x_2 &= k_1 \\ x_2 + x_3 &= k_2 \\ 3x_1 - x_2 - 14x_3 &= k_3 \end{aligned}$$ To express this system as a matrix equation, we write it in the form: $$A \mathbf{x} = \mathbf{k}$$ where: $$A = \begin{pmatrix} 1 & 4 & 0 \\ 0 & 1 & 1 \\ 3 & -1 & -14 \end{pmatrix}, \quad \mathbf{x} = \begin{pmatrix} x_1 \\ x_2 \\ x_3 \end{pmatrix}, \quad \mathbf{k} = \begin{pmatrix} k_1 \\ k_2 \\ k_3 \end{pmatrix}$$ ### Part a: Write the matrix equation The matrix equation is: $$\begin{pmatrix} 1 & 4 & 0 \\ 0 & 1 & 1 \\ 3 & -1 & -14 \end{pmatrix} \begin{pmatrix} x_1 \\ x_2 \\ x_3 \end{pmatrix} = \begin{pmatrix} k_1 \\ k_2 \\ k_3 \end{pmatrix}$$ ### Part b: Solve using matrix inverses We solve for $$\mathbf{x}$$ using the matrix inverse: $$\mathbf{x} = A^{-1} \mathbf{k}$$ To find $$A^{-1}$$, we calculate the inverse of the matrix $$A$$. The inverse of a 3x3 matrix can be found using the formula for the inverse of a general matrix, or by row reduction. For simplicity, let's proceed directly with the formula. Now, given the specific values of $$k_1 = 32$$, $$k_2 = -3$$, and $$k_3 = 142$$, we substitute these into the equation: $$\mathbf{k} = \begin{pmatrix} 32 \\ -3 \\ 142 \end{pmatrix}$$ After finding $$A^{-1}$$, we multiply it by $$\mathbf{k}$$ to obtain $$\mathbf{x}$$, which consists of $$x_1$$, $$x_2$$, and $$x_3$$. Let me calculate the inverse of $$A$$ and solve for $$\mathbf{x}$$. The solution to the system of equations is: $$x_1 = 16, \quad x_2 = 4, \quad x_3 = -7$$ ### Part c: Conclusion For the given values $$k_1 = 32$$, $$k_2 = -3$$, and $$k_3 = 142$$, the values of $$x_1$$, $$x_2$$, and $$x_3$$ are: $$x_1 = 16, \quad x_2 = 4, \quad x_3 = -7$$ Would you like more details on the steps, or do you have any questions? Here are 5 related questions to expand on this topic: 1. How do you find the inverse of a 3x3 matrix step by step? 2. What are the conditions for a matrix to have an inverse? 3. Can you solve this system using another method, such as Gaussian elimination? 4. How does the determinant of a matrix relate to the existence of an inverse? 5. What happens if the matrix $$A$$ is singular and has no inverse? **Tip**: When solving systems of equations using matrices, ensure the matrix is invertible by checking that its determinant is non-zero.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Linear Algebra
Systems of Equations
Matrix Inverses
Formulas
Matrix equation: A * x = k
Matrix inverse: A^(-1)
Multiplication of matrices
Theorems
Inverse Matrix Theorem
Suitable Grade Level
College Level (Undergraduate)