91Ó°ÊÓ

Matrix-vector multiplication is a fundamental operation in linear algebra, and it is crucial for understanding how systems of equations function in the context of differential equations. In our problem, we have a matrix \( A \) and a vector \( \mathbf{X} \). The goal is to multiply \( A \) by \( \mathbf{X} \).To do this, follow these steps:

• Identify the rows of the matrix \( A \).
• Multiply each element of a matrix row by the corresponding element of the vector.
• Sum the products for each row.

In our example, \( A \) is a \( 2 \times 2 \) matrix and \( \mathbf{X} \) is a 2-dimensional vector. Thus, the resulting product will itself be a vector. Specifically, that's how we arrived at\[A\mathbf{X} = \begin{pmatrix} (1 + t)e^{-wh} \ -3e^{-wh} \end{pmatrix}.\]Understanding this process helps determine the correctness of \( \mathbf{X} \) as a solution to the differential equation system.

Verifying whether a vector \( \mathbf{X} \) is indeed a solution to a given system of differential equations is a typical task. It ensures that both sides of the differential equation—from the derivative of \( \mathbf{X} \) to the matrix-vector product—are consistent.Begin by calculating the derivative of \( \mathbf{X} \), as done in our step-by-step solution. Here, utilizing the product rule allows computation of this derivative accurately.Next, compare this derivative with the calculated matrix-vector product. Both expressions should match. If they don't, as in our example, \( \mathbf{X} \) cannot be considered a solution without further adjustments or assumptions. Think of this verification step as a consistency check in mathematical terms.

A system of equations consists of multiple equations that share the same variables. Each equation represents a different constraint, or rule, that the solution must satisfy. In the context of differential equations, these systems often model real-world dynamic processes.When working with systems of differential equations, emphasize:

• The relationship between each equation and how variables interact.
• The method of solving, such as eigenvalues for linear systems or numerical methods for more complex ones.

For our problem, the system is represented by a matrix equation \( \mathbf{X}' = A\mathbf{X} \), where we're tasked to check if a candidate \( \mathbf{X} \) satisfies this relationship across every equation. Ensuring each condition holds provides a complete solution to the entire system.

The product rule is a fundamental tool in calculus for finding the derivative of a product of two functions. Mathematically, if \( f(t) \) and \( g(t) \) are both functions of \( t \), the product rule states:\[( f \cdot g )' = f' \cdot g + f \cdot g'\]In our exercise, \( \mathbf{X} \) is notably expressed as a scalar product: a constant vector times an exponential function, \( e^{-wh} \). Applying the product rule:

• Differentiates the vector function separately from the exponential.
• Leads to calculating \( f'(t)\cdot g(t) + f(t) \cdot g'(t) \).

This method resulted in us finding the derivative \( \mathbf{X}' \), necessary for verifying it against the matrix product side of the equation. The product rule simplifies complex systems into simpler, manageable parts for differentiation.