Problem 15 - Advanced Engineering Mathematics

Question

In Problems 11-16, verify that the vector \(\mathbf{X}\) is a solution of the given system. $$ \mathbf{X}^{\prime}=\left(\begin{array}{rrr} 1 & 2 & 1 \\ 6 & -1 & 0 \\ -1 & -2 & -1 \end{array}\right) \mathbf{X} ; \quad \mathbf{X}=\left(\begin{array}{r} 1 \\ 6 \\ -13 \end{array}\right) $$

Step-by-Step Solution

Answer

\( \mathbf{X} \) is a solution since \( A\mathbf{X} = \mathbf{X}' = \begin{pmatrix} 0 \\ 0 \\ 0 \end{pmatrix} \)."

Step 1: Understand the Problem

We need to verify if the vector \( \mathbf{X} = \begin{pmatrix} 1 \ 6 \ -13 \end{pmatrix} \) is a solution to the system \( \mathbf{X}' = A\mathbf{X} \), where the matrix \( A \) is \( \begin{pmatrix} 1 & 2 & 1 \ 6 & -1 & 0 \ -1 & -2 & -1 \end{pmatrix} \). This means we need to check if substituting \( \mathbf{X} \) into the equation \( A\mathbf{X} \) gives us the same result for \( \mathbf{X}' \).

Step 2: Perform Matrix-Vector Multiplication

Multiply the matrix \( A \) by the vector \( \mathbf{X} \): \[ A\mathbf{X} = \begin{pmatrix} 1 & 2 & 1 \ 6 & -1 & 0 \ -1 & -2 & -1 \end{pmatrix} \begin{pmatrix} 1 \ 6 \ -13 \end{pmatrix} \] Perform the product for each row of the matrix and the vector.

Step 3: Calculate Each Component of A\(\mathbf{X}\)

First component: \[ 1(1) + 2(6) + 1(-13) = 1 + 12 - 13 = 0 \] Second component: \[ 6(1) + (-1)(6) + 0(-13) = 6 - 6 + 0 = 0 \] Third component: \[ -1(1) + (-2)(6) + (-1)(-13) = -1 - 12 + 13 = 0 \] Thus, \( A\mathbf{X} = \begin{pmatrix} 0 \ 0 \ 0 \end{pmatrix} \).

Step 4: Verification of Solution

The result of matrix-vector multiplication is \( \begin{pmatrix} 0 \ 0 \ 0 \end{pmatrix} \), which is consistent with \( \mathbf{X}' = \begin{pmatrix} 0 \ 0 \ 0 \end{pmatrix} \), indicating that \( \mathbf{X} \) is indeed a solution.

Key Concepts

Matrix-Vector Multiplication

Matrix-vector multiplication is a fundamental concept in linear algebra. It involves multiplying a matrix by a vector. In this context, the matrix is a collection of numbers arranged in rows and columns, and the vector is a single-column collection of numbers. Multiplying a matrix by a vector involves a series of dot products between each row of the matrix and the vector.
In the exercise above, we multiply matrix \( A \) by vector \( \mathbf{X} \). For each element of the resulting vector, we compute the dot product of the corresponding row of the matrix with the vector. For example, for the first component, we compute \( 1 \times 1 + 2 \times 6 + 1 \times (-13) \). This process is repeated for each row to give the entire result of the multiplication.
As a result, matrix-vector multiplication transforms the original vector \( \mathbf{X} \) into a new vector based on the operations defined by the matrix.

Solution Verification

Solution verification involves confirming that a proposed solution satisfies the given mathematical equations or system. In differential equations, we verify solutions by substituting them back into the original equation. If both sides of the equation match, we can be confident the solution is correct.
In the example above, the solution verification requires checking that \( A\mathbf{X} \) matches \( \mathbf{X}' \). We've multiplied the matrix by the vector to get \( \begin{pmatrix} 0 \ 0 \ 0 \end{pmatrix} \), which matches \( \mathbf{X}' \). This match indicates the provided vector \( \mathbf{X} \) is indeed a solution to the system.

• Check both sides of the equation for equality.
• Ensure any calculations align with the expected result.

Linear Algebra

Linear algebra is a branch of mathematics dealing with vectors, matrices, and the linearly related systems they form. It's a crucial tool for solving systems of equations, especially when these systems involve multiple variables and equations.
In the context of this exercise, linear algebra provides the framework and tools necessary to understand and solve the system of differential equations. The matrix \( A \) and vector \( \mathbf{X} \) are both linear algebra elements that are manipulated to verify a solution.
Linear algebra helps us understand the relationships between variables and the transformations a matrix can apply to a vector. This understanding is foundational for many scientific and engineering disciplines.

• Vectors represent quantities with magnitude and direction.
• Matrices represent linear transformations and can solve systems of linear equations.
• Algebraic operations between vectors and matrices, such as addition and multiplication, work according to specific rules in linear algebra.

Differential Equations

Differential equations incorporate derivatives, allowing us to model relationships involving rates of change. They are pervasive in scientific fields because real-world systems often depend on how things change over time.
Systems of linear differential equations involve derivatives of several interrelated functions. In these systems, each function's rate of change may depend on multiple variables. In our worked example, the vector function \( \mathbf{X} \) represents the solution to the differential system, and \( \mathbf{X}' = A\mathbf{X} \) is the system's differential equation.
Solving a system like this involves finding a vector \( \mathbf{X} \) such that its derivative matches the transformation applied by the matrix, thus modeling the dynamic changes dictated by the system of equations.

• Differential equations describe continuously changing processes.
• Solutions may include functions that meet specific conditions outlined by the equations.
• Systems of differential equations can represent multiple, interdependent changes.