/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 49 Use an inverse matrix to solve t... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 8: Problem 49

Use an inverse matrix to solve the given system of equations. $$ \begin{gathered} x_{1}+2 x_{2}+2 x_{3}=1 \\ x_{1}-2 x_{2}+2 x_{3}=-3 \\ 3 x_{1}-x_{2}+5 x_{3}=7 \end{gathered} $$

Short Answer

Expert verified
\( x_1 = 2, x_2 = 1, x_3 = 0 \)

Step by step solution

01

Write the system of equations as a matrix equation

First, represent the system of equations in matrix form: \( AX = B \), where: \[ A = \begin{bmatrix} 1 & 2 & 2 \ 1 & -2 & 2 \ 3 & -1 & 5 \end{bmatrix}, \, X = \begin{bmatrix} x_1 \ x_2 \ x_3 \end{bmatrix}, \, B = \begin{bmatrix} 1 \ -3 \ 7 \end{bmatrix} \].
02

Find the inverse of matrix A

Calculate the inverse of matrix \( A \) using the formula for finding the inverse of a 3x3 matrix. If \( A = \begin{bmatrix} a & b & c \ d & e & f \ g & h & i \end{bmatrix} \), then \[ A^{-1} = \frac{1}{\text{det}(A)} \begin{bmatrix} ei-fh & ch-bi & bf-ce \ fg-di & ai-cg & cd-af \ dh-eg & bg-ah & ae-bd \end{bmatrix} \].
03

Calculate the determinant of A

The determinant of \( A \) is calculated as follows. \[ \text{det}(A) = 1(5 \cdot -2 - (-1) \cdot 2) - 2(1 \cdot 5 - 3 \cdot 2) + 2(1 \cdot (-1) - 3\cdot -2) \]. Calculate this to get the determinant value.
04

Compute the inverse of A

Using the determinant from the previous step, compute \[ A^{-1} \]. Simplify each entry using the determinant to divide the cofactor matrix from Step 2.
05

Solve for X using A^{-1}B

Multiply \( A^{-1} \) by \( B \) to find \( X \). \[ X = A^{-1}B \]Perform the matrix multiplication.
06

Identify the solution to the system

The resulting matrix \( X \) will have the solution. If \( X = \begin{bmatrix} x_1 \ x_2 \ x_3 \end{bmatrix} \), the solution is the values for \( x_1, x_2, \) and \( x_3 \) obtained from the matrix in Step 5.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Matrix Equation
A matrix equation is a way to represent a system of linear equations using matrices. Imagine you have several equations with multiple variables. Instead of dealing with each one individually, you can bundle them into a compact and structured form. In mathematical terms, you write it as \( AX = B \), where:
  • \( A \) is a matrix containing coefficients from each equation.
  • \( X \) is a column matrix representing the variables you're solving for, like \( x_1, x_2, \) and \( x_3 \).
  • \( B \) is another column matrix with the constants from the right side of each equation.
This form makes it easier to handle and solve using various matrix operations.
Determinant of a Matrix
The determinant is a special number that can be calculated from a square matrix. It is crucial because it tells us if a matrix has an inverse. For a 3x3 matrix \( A \), the determinant is calculated using a formula involving the elements of \( A \). To check if a matrix is invertible, its determinant should not be zero. The formula for calculating the determinant of a 3x3 matrix \( A = \begin{bmatrix} a & b & c \ d & e & f \ g & h & i \end{bmatrix} \) is:\[\text{det}(A) = a(ei - fh) - b(di - fg) + c(dh - eg)\]If \( \text{det}(A) eq 0 \), the inverse exists, and the matrix can be used in further operations like solving matrix equations.
Matrix Multiplication
Matrix multiplication is an operation where you multiply two matrices and get a new matrix. When using matrix multiplication to solve equations, it's crucial to follow the correct order. To multiply matrix \( A \) with matrix \( B \), the number of columns in \( A \) must match the number of rows in \( B \). This condition ensures each element of the resulting matrix is computed accurately. The key steps include:
  • Take a row from the first matrix and a column from the second matrix.
  • Multiply corresponding elements and add the products.
  • Repeat this for each row of the first matrix and each column of the second matrix.
In the context of solving equations, you might see something like \( X = A^{-1}B \). Here, you multiply the inverse of matrix \( A \) with matrix \( B \) to find the solution matrix \( X \).
3x3 Matrices
A 3x3 matrix is a grid with 3 rows and 3 columns, making it square. Each element in the matrix can represent different coefficients or constants in mathematical problems. Understanding a 3x3 matrix requires recognizing how these nine elements can interact. Key points to consider include:
  • The interaction of rows (horizontal arrays) and columns (vertical arrays).
  • Operations like addition, subtraction, determinant calculation, and finding inverses require treating these matrices with systematic methods.
  • 3x3 matrices are often more complex than smaller matrices but are still manageable with these systematic techniques.
Using a 3x3 matrix is common in solving systems of equations with three variables, and understanding the properties of these matrices is integral in higher-level mathematics.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Use an inverse matrix to solve the given system of equations. $$ \begin{aligned} x_{1}-\quad x_{3}-2 &=2 \\ x_{2}+x_{3} &=1 \\ -x_{1}+x_{2}+2 x_{3}+x_{4} &=-5 \\ x_{3}-x_{4} &=3 \end{aligned} $$

Find the least squares line for the given data. $$ (0,-1),(1,3),(2,5),(3,7) $$

Determine whether the given message is a code word in the Hamming \((7,4)\) code. If it is, decode it. If it is not, correct the single error and decode the corrected message. $$ \left.\begin{array}{lllllll} (0 & 1 & 1 & 1 & 0 & 0 & 1 \end{array}\right) $$

Determine whether the given message is a code word in the Hamming \((7,4)\) code. If it is, decode it. If it is not, correct the single error and decode the corrected message. $$ \left.\begin{array}{lllllll} (1 & 0 & 0 & 1 & 0 & 0 & 1 \end{array}\right) $$

(a) verify that the indicated column vectors are eigenvectors of the given symmetric matrix and (b) identify the corresponding eigenvalues. (c) Proceed as in Example 4 and use the Gram-Schmidt process to construct an orthogonal matrix \(\mathbf{P}\) from the eigenvectors. $$ \begin{aligned} &\mathbf{A}=\left(\begin{array}{lll} 0 & 2 & 2 \\ 2 & 0 & 2 \\ 2 & 2 & 0 \end{array}\right) ; \quad \mathbf{K}_{1}=\left(\begin{array}{r} 1 \\ -1 \\ 0 \end{array}\right), \\ &\mathbf{K}_{2}=\left(\begin{array}{r} 1 \\ 0 \\ -1 \end{array}\right), \quad \mathbf{K}_{3}=\left(\begin{array}{l} 1 \\ 1 \\ 1 \end{array}\right) \end{aligned} $$
See all solutions

Recommended explanations on Physics Textbooks

Electric Charge Field and Potential

Read Explanation

Solid State Physics

Read Explanation

Particle Model of Matter

Read Explanation

Dynamics

Read Explanation

Magnetism and Electromagnetic Induction

Read Explanation

Atoms and Radioactivity

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.