Problem 5 Express the following sets of si... [FREE SOLUTION]

Chapter 17: Problem 5

Express the following sets of simultaneous equations in matrix form: (a) $$ \begin{array}{rl} 2 x_{1}-3 x_{2}-x_{3}=2 & \text { (b) } x_{1}-2 x_{2}-x_{3}+3 x_{4}=10 \\ x_{1}+4 x_{2}+2 x_{3}=3 & 2 x_{1}+3 x_{2}+x_{4}=8 \\ x_{1}-x_{2}+x_{3}=5 & x_{1}-4 x_{3}-2 x_{4}=3 \\ & -x_{2}+3 x_{3}+x_{4}=-7 \end{array} $$

Short Answer

Expert verified

The first system is $AX = B$ and the second system is $CY = D$.

Step by step solution

- Write the given equations in order

Identify the equations given in each set and align them.

- Write coefficient matrices

Extract coefficients of each variable for both systems and organize them into matrices. The first system's coefficient matrix is: \[ A = \begin{pmatrix} 2 & -3 & -1 \ 1 & 4 & 2 \ 1 & -1 & 1 \end{pmatrix} \]

- Write variable and constant matrices for the first system

The variable matrix for the first system is: \[ X = \begin{pmatrix} x_1 \ x_2 \ x_3 \end{pmatrix} \] The constant matrix for the first system is: \[ B = \begin{pmatrix} 2 \ 3 \ 5 \end{pmatrix} \]

- Combine matrices into one equation for the first system

Express the first system of equations in matrix form: \[ AX = B \] Where \[ \begin{pmatrix} 2 & -3 & -1 \ 1 & 4 & 2 \ 1 & -1 & 1 \end{pmatrix} \begin{pmatrix} x_1 \ x_2 \ x_3 \end{pmatrix} = \begin{pmatrix} 2 \ 3 \ 5 \end{pmatrix} \]

- Write coefficient matrices for the second system

Extract coefficients for the second system and organize them into matrices. The second system's coefficient matrix is: \[ C = \begin{pmatrix} 1 & -2 & -1 & 3 \ 2 & 3 & 0 & 1 \ 1 & 0 & -4 & -2 \ 0 & -1 & 3 & 1 \end{pmatrix} \]

- Write variable and constant matrices for the second system

The variable matrix for the second system is: \[ Y = \begin{pmatrix} x_1 \ x_2 \ x_3 \ x_4 \end{pmatrix} \] The constant matrix for the second system is: \[ D = \begin{pmatrix} 10 \ 8 \ 3 \ -7 \end{pmatrix} \]

- Combine matrices into one equation for the second system

Express the second system of equations in matrix form: \[ CY = D \] Where \[ \begin{pmatrix} 1 & -2 & -1 & 3 \ 2 & 3 & 0 & 1 \ 1 & 0 & -4 & -2 \ 0 & -1 & 3 & 1 \end{pmatrix} \begin{pmatrix} x_1 \ x_2 \ x_3 \ x_4 \end{pmatrix} = \begin{pmatrix} 10 \ 8 \ 3 \ -7 \end{pmatrix} \]

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.

linear algebra

Linear algebra is a branch of mathematics that studies vectors, vector spaces, and linear equations. It provides powerful tools for matrix operations, which are especially useful in solving simultaneous equations. When dealing with multiple equations, linear algebra allows us to simplify and solve them systematically using matrices. Matrices facilitate a more compact and efficient way to handle large systems of equations. This transformation from equations to matrix form not only makes the process cleaner but also opens the door to employing numerous computational techniques.

coefficient matrix

The coefficient matrix, also known as the matrix of coefficients, contains the coefficients of the variables from the set of equations. In our example, for the first system, the coefficient matrix is noted as $ A = \begin{pmatrix} 2 & -3 & -1 \ 1 & 4 & 2 \ 1 & -1 & 1 \ \text{end{pmatrix}$. This matrix exclusively shows the numerical values multiplying the variables in the equations. Constructing the coefficient matrix is one of the initial and crucial steps in transforming a set of simultaneous equations into matrix form, as it identifies and organizes all the relevant coefficients.

variable matrix

The variable matrix is a column matrix that includes the variables of the given system of equations. For instance, in the first system, the variable matrix is $ X = \begin{pmatrix} x_1 \ x_2 \ x_3 \ \text{end{pmatrix}$. It鈥檚 essential because it isolates the variables that we need to solve for. Each element in the variable matrix corresponds to a variable in the equations. Thus, when multiplied by the coefficient matrix, it replicates the equation structure in a compact form, allowing us to solve for these unknowns systematically.

constant matrix

The constant matrix, also called the matrix of constants, contains the constants from the right-hand side of the equations. For example, the first system鈥檚 constant matrix is $ B = \begin{pmatrix} 2 \ 3 \ 5 \ \text{end{pmatrix}$. Much like the coefficient matrix and variable matrix, the constant matrix plays a crucial role in expressing the equations in matrix form. It holds the values that result from the linear combinations of variables and their respective coefficients, providing the necessary balance in the equation $ AX = B $. The constant matrix ensures that the system鈥檚 equations are accurately represented when you combine them into a singular matrix equation.

91影视

Short Answer

Step by step solution

- Write the given equations in order

- Write coefficient matrices

- Write variable and constant matrices for the first system

- Combine matrices into one equation for the first system

- Write coefficient matrices for the second system

- Write variable and constant matrices for the second system

- Combine matrices into one equation for the second system

Key Concepts

linear algebra

coefficient matrix

variable matrix

constant matrix

One App. One Place for Learning.

Recommended explanations on Math Textbooks

Pure Maths

Mechanics Maths

Geometry

Calculus

Statistics

Probability and Statistics

Study anywhere. Anytime. Across all devices.