write given system in matrix form x'=Ax+f dx/dt=x+y+z dy/dt=7x-y-2z dz/dt=x-2z

To write the given system of differential equations in matrix form \( \mathbf{x}' = A\mathbf{x} + \mathbf{f} \), we start by expressing the system in terms of vector and matrix notation.

The given system is:
\[
\begin{cases}
\frac{dx}{dt} = x + y + z \\
\frac{dy}{dt} = 7x - y - 2z \\
\frac{dz}{dt} = x - 2z
\end{cases}
\]

First, we define the vector \(\mathbf{x}\) and its derivative \(\mathbf{x}'\):
\[
\mathbf{x} = \begin{pmatrix}
x \\
y \\
z
\end{pmatrix}, \quad \mathbf{x}' = \frac{d}{dt}\begin{pmatrix}
x \\
y \\
z
\end{pmatrix} = \begin{pmatrix}
\frac{dx}{dt} \\
\frac{dy}{dt} \\
\frac{dz}{dt}
\end{pmatrix}
\]

Next, we write the system in matrix form by identifying the coefficient matrix \(A\) and the vector \(\mathbf{f}\). Here, \(\mathbf{f}\) is a zero vector since there are no constant terms in the system.

\[
\mathbf{x}' = \begin{pmatrix}
\frac{dx}{dt} \\
\frac{dy}{dt} \\
\frac{dz}{dt}
\end{pmatrix} = \begin{pmatrix}
x + y + z \\
7x - y - 2z \\
x - 2z
\end{pmatrix}
\]

We can represent this as:
\[
\mathbf{x}' = A\mathbf{x} + \mathbf{f}
\]

Where:
\[
A = \begin{pmatrix}
1 & 1 & 1 \\
7 & -1 & -2 \\
1 & 0 & -2
\end{pmatrix}, \quad \mathbf{f} = \begin{pmatrix}
0 \\
0 \\
0
\end{pmatrix}
\]

Thus, the matrix form of the given system is:
\[
\mathbf{x}' = \begin{pmatrix}
1 & 1 & 1 \\
7 & -1 & -2 \\
1 & 0 & -2
\end{pmatrix} \begin{pmatrix}
x \\
y \\
z
\end{pmatrix} + \begin{pmatrix}
0 \\
0 \\
0
\end{pmatrix}
\]

Would you like more details or have any questions?

Here are 8 related questions that might help you expand your understanding:

1. How can you find the eigenvalues of the matrix \(A\)?
2. What is the significance of the eigenvalues and eigenvectors in solving the system?
3. How do you solve the homogeneous system \(\mathbf{x}' = A\mathbf{x}\)?
4. What is the general solution for a system of linear differential equations?
5. How can you use the matrix exponential \(e^{At}\) to solve the system?
6. What happens if the system has non-zero constant terms in \(\mathbf{f}\)?
7. How do you determine if the system is stable?
8. How can you transform a higher-order differential equation into a system of first-order differential equations?

**Tip:** To solve a system of differential equations, finding the eigenvalues and eigenvectors of the coefficient matrix \(A\) is crucial as they provide insight into the system's behavior and solutions.

Understand how to express a system of differential equations in matrix form x' = Ax + f. This problem involves converting differential equations into vector notation and identifying the coefficient matrix A and vector f. Suitable for advanced high school or college-level mathematics.

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