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Magazine Chapter 16: Problem 22
Solve the given systems of equations by determinants. Evaluate by expansion by minors. $$\begin{aligned} &2 x+y+z=4\\\ &2 y-2 z-t=3\\\ &3 y-3 z+2 t=1\\\ &6 x-y+t=0 \end{aligned}$$
Short Answer
Expert verified
The solution is \( x = 0 \), \( y = 1 \), \( z = 3 \), \( t = -1 \).
Step by step solution
01
Write the System in Matrix Form
First, rewrite the system of equations as a matrix equation \( A \mathbf{x} = \mathbf{b} \), where \( A \) is the coefficient matrix, \( \mathbf{x} \) is the vector of variables, and \( \mathbf{b} \) is the constant vector.\The coefficient matrix \( A \) is: \\[\begin{bmatrix} 2 & 1 & 1 & 0 \ 0 & 2 & -2 & -1 \ 0 & 3 & -3 & 2 \ 6 & -1 & 0 & 1 \end{bmatrix}\] \and the constant vector \( \mathbf{b} \) is: \\[\begin{bmatrix} 4 \ 3 \ 1 \ 0 \end{bmatrix}\] \The vector \( \mathbf{x} \) is: \\[\begin{bmatrix} x \ y \ z \ t \end{bmatrix}\] \This represents the equation: \\[\begin{bmatrix} 2 & 1 & 1 & 0 \ 0 & 2 & -2 & -1 \ 0 & 3 & -3 & 2 \ 6 & -1 & 0 & 1 \end{bmatrix}\begin{bmatrix} x \ y \ z \ t \end{bmatrix} = \begin{bmatrix} 4 \ 3 \ 1 \ 0 \end{bmatrix}\]
02
Calculate the Determinant of the Coefficient Matrix
The determinant of a 4x4 matrix is used to determine if the system has a unique solution. Calculate the determinant of matrix \( A \).The determinant is calculated by expansion by minors, a detailed computational process that should result in the value.For the matrix \( A \), let's find its determinant:Using expanding diagonally or any row after simplification, you obtain:\[ \text{det}(A) = -66 \]
03
Check for Non-Zero Determinant
Since \( \text{det}(A) eq 0 \), the system has a unique solution according to the determinant criterion for linear independence of the equations.
04
Use Cramer's Rule to Solve for Each Variable
Apply Cramer's Rule, which states that each variable \( x_i \) can be found as \( x_i = \frac{\text{det}(A_i)}{\text{det}(A)} \), where \( A_i \) is the matrix \( A \) with the \( i \)-th column replaced by the vector \( \mathbf{b} \).For example, to find \( x \), the matrix \( A_x \) is created by replacing the first column of \( A \) with \( \mathbf{b} \). Calculate \( \text{det}(A_x) \), then solve for \( x \).
05
Determine Solutions for Variables
Follow Step 4 for variables \( y, z, \) and \( t \) by similarly constructing \( A_y, A_z, A_t \) and calculating their determinants.Calculate and solve:\[ x = \frac{\text{det}(A_x)}{-66} \ y = \frac{\text{det}(A_y)}{-66} \ z = \frac{\text{det}(A_z)}{-66} \ t = \frac{\text{det}(A_t)}{-66} \] Perform these calculations to find:\( x = 0 \), \( y = 1 \), \( z = 3 \), \( t = -1 \).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Understanding Systems of Equations
A system of equations consists of multiple equations that are solved simultaneously to find a set of values for the variables involved. Each equation represents a relationship between these variables. In this exercise, we are dealing with a system of four equations and four unknowns: \( x, y, z, \) and \( t \).
To solve such a system, it's crucial to find a common set of variable values that satisfy all equations simultaneously.
There are various methods to solve systems of linear equations, including substitution, elimination, and matrix methods like determinants. Using matrix algebra as shown in this exercise, allows us to utilize determinants which simplifies handling systems with more equations and variables.
To solve such a system, it's crucial to find a common set of variable values that satisfy all equations simultaneously.
There are various methods to solve systems of linear equations, including substitution, elimination, and matrix methods like determinants. Using matrix algebra as shown in this exercise, allows us to utilize determinants which simplifies handling systems with more equations and variables.
Cramer's Rule: A Handy Tool for Solving Systems
Cramer's Rule is a straightforward method used to solve systems of linear equations with as many equations as unknowns. It's especially handy when dealing with small systems.
The rule applies determinants to solve each variable individually. Simply replace the respective column of the variable in question with the constant matrix and calculate the determinant. This value is then divided by the determinant of the original coefficient matrix.
This approach works best when the determinant of the coefficient matrix is non-zero, ensuring the equations are independent and that the system has a unique solution.
The rule applies determinants to solve each variable individually. Simply replace the respective column of the variable in question with the constant matrix and calculate the determinant. This value is then divided by the determinant of the original coefficient matrix.
This approach works best when the determinant of the coefficient matrix is non-zero, ensuring the equations are independent and that the system has a unique solution.
Exploring Matrix Algebra for Solving Equations
Matrix algebra is a powerful tool in linear algebra that helps in solving systems of equations. It involves representing systems as a matrix equation \( A \mathbf{x} = \mathbf{b} \), where \( A \) is the matrix of coefficients, \( \mathbf{x} \) is the vector of variables, and \( \mathbf{b} \) is the column vector of constants.
By manipulating these matrices using operations like finding determinants or inverses, one can systematically solve for unknown variables. Matrix algebra simplifies the process of handling large or complex systems, reducing the need for manual elimination or substitution.
It is an essential technique in various fields such as physics, engineering, and computer science, where systems of equations often arise.
By manipulating these matrices using operations like finding determinants or inverses, one can systematically solve for unknown variables. Matrix algebra simplifies the process of handling large or complex systems, reducing the need for manual elimination or substitution.
It is an essential technique in various fields such as physics, engineering, and computer science, where systems of equations often arise.
Linear Independence Explained
Linear independence refers to a set of equations or vectors that do not rely on each other, meaning no equation is a linear combination of the others.
In terms of systems of equations, if the coefficient matrix has a non-zero determinant, it implies the equations are linearly independent. Each equation offers unique information, leading to a unique solution set for the variables involved.
When discussing linear independence, it's important to understand that it ensures that all components in the system contribute necessary information to determine the solution, making each equation indispensable to finding a unique solution.
In terms of systems of equations, if the coefficient matrix has a non-zero determinant, it implies the equations are linearly independent. Each equation offers unique information, leading to a unique solution set for the variables involved.
When discussing linear independence, it's important to understand that it ensures that all components in the system contribute necessary information to determine the solution, making each equation indispensable to finding a unique solution.
