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Chapter 13: Problem 18

Find all of the solutions of the systems. $$ \left(\begin{array}{rr} 1 & 4 \\ -1 & 1 \end{array}\right)\left(\begin{array}{l} x \\ y \end{array}\right)=0 $$

Short Answer

Expert verified
The solution is \( x = 0 \) and \( y = 0 \).

Step by step solution

01

Set Up the System of Equations

The given matrix equation can be expanded into two separate linear equations. From the matrix multiplication, \( \begin{pmatrix} 1 & 4 \ -1 & 1 \end{pmatrix} \begin{pmatrix} x \ y \end{pmatrix} = \begin{pmatrix} 0 \ 0 \end{pmatrix} \), we derive the following system:1. \( 1x + 4y = 0 \).2. \( -1x + 1y = 0 \).
02

Solve the First Equation

From the first equation, \( x + 4y = 0 \), solve for \( x \). This gives:\[ x = -4y \].
03

Solve the Second Equation using the First Equation's Solution

Substitute \( x = -4y \) from Step 2 into the second equation \( -x + y = 0 \):\[ -(-4y) + y = 0 \].This simplifies to \( 4y + y = 0 \), which further simplifies to \( 5y = 0 \).
04

Solve for \( y \)

From \( 5y = 0 \), we find that \( y = 0 \).
05

Determine \( x \) Using the Value of \( y \)

Substitute the value of \( y = 0 \) back into the equation \( x = -4y \):\[ x = -4(0) = 0 \].
06

Conclusion of Solutions

The solution to the system is \( x = 0 \) and \( y = 0 \), or simply \( \begin{pmatrix} 0 \ 0 \end{pmatrix} \). This means the only solution is the trivial solution where both variables are zero.

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Key Concepts

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

Matrix Equations
Matrix equations are a key part of linear algebra. They involve the multiplication of matrices and vectors to find solutions. In simple terms:
  • The matrix provides coefficients for variables.
  • The vector represents the variables.
  • Multiplying them gives the result vector.
In the exercise, the matrix equation involves a matrix multiplied by a vector equaling zero. The general form of such a matrix equation is:\[ A\mathbf{x} = \mathbf{0} \]where \( A \) is the matrix of coefficients, and \( \mathbf{x} \) is the vector of variables.To solve:
  • Extract linear equations by examining each row of the matrix.
  • This converts the matrix equation into individual linear equations for the variables \( x \) and \( y \).
Understanding matrix equations helps to solve systems of equations efficiently, especially with larger systems.
Linear Systems
A linear system involves two or more linear equations that are connected by shared variables. In the context of the exercise, you are dealing with the linear system resulting from:\[ \begin{aligned} &1x + 4y = 0 \ &-1x + 1y = 0 \end{aligned} \]To solve a linear system, especially when tackling multiple equations, the goal is to find values for the variables that satisfy all equations simultaneously. Here鈥檚 how you can approach solving a linear system:
  • Substitute: Isolate a variable in one equation, then substitute into the other.
  • Elimination: Add or subtract equations to eliminate a variable, simplifying the system.
  • Check: Always verify solutions by substituting back into the original equations to ensure all are satisfied.
By solving the given linear system, you find the intersection point or where the lines represented by the equations meet.
Trivial Solution
In linear algebra, a trivial solution occurs when the only solution to the system is for all variables to be zero. For the provided matrix equation, we discovered that:\[ x = 0 \quad \text{and} \quad y = 0 \]This is a classic case of a trivial solution. Here are some characteristics of trivial solutions:
  • They denote the simplest solution.
  • Such a result in a homogeneous system (a system where all constants are zero) is common.
  • Finding only a trivial solution usually implies the system does not describe an enriched geometry like an intersection point outside the origin.
Understanding the concept of a trivial solution is key, as it forms the basis for recognizing when a system of equations has a straightforward resolution, often with no additional insights beyond the origin.

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Most popular questions from this chapter

Write the given system of differential equations as a matrix equation. $$ \begin{array}{l} \frac{d x}{d t}=t x+y+\sin t \\ \frac{d y}{d t}=t^{2} x+t y+1 \end{array} $$

Prove \(D+E=E+D\) for any matrices \(D\) and \(E\) for which the sum is defined.

Solve the system \(X^{\prime}=A X\). $$ A=\left(\begin{array}{rrr} 2 & 1 & -1 \\ 0 & -1 & 2 \\ 0 & 0 & -1 \end{array}\right) $$

Consider the initial value problem $$ \begin{array}{l} x^{\prime}(t)+2 y(t)=0 \\ x^{\prime \prime}(t)+2 y^{\prime}(t)+2 y(t)=2 e^{t} \\ x(0)=1, x^{\prime}(0)=0, y(0)=0 \end{array} $$ (a) Show that the Laplace transform method produces \(x=3-2 e^{t}, \quad y=e^{t}\) (b) Verify that these functions satisfy the differential equations but do not satisfy the initial conditions. (c) By elementary elimination, show that a solution of the system of differential equations has the form \(x=c_{1}-2 e^{t}, \quad y=e^{t}\)

Replace the given equation by a system of first order equations. $$ y^{\prime \prime}+p y^{\prime}+q y=f(x) $$
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