https://file.aiquickdraw.com/mathbot/file/1725591264506phf222df.jpg

The equation provided in the image is:

\[
\theta_{1}^{T} a(s) P(s) + (\theta_{2}^{T} a(s) + \theta_{0}^{T} a(s)) \lambda_{s} Z(s) = A(s) P(s) - k_{\rho} Z(s) P_{\rho} (s)
\]

To solve for \(\theta_1\), \(\theta_2\), and \(\theta_0\), we need to separate these terms and express them in terms of other variables. Here's a step-by-step outline:

1. **Combine terms involving the thetas** on the left-hand side:

   \[
   \theta_{1}^{T} a(s) P(s) + (\theta_{2}^{T} a(s) + \theta_{0}^{T} a(s)) \lambda_{s} Z(s)
   \]

   This simplifies to:

   \[
   \theta_{1}^{T} a(s) P(s) + \theta_{2}^{T} a(s) \lambda_{s} Z(s) + \theta_{0}^{T} a(s) \lambda_{s} Z(s)
   \]

   which can be grouped as:

   \[
   \theta_{1}^{T} a(s) P(s) + (\theta_{2}^{T} + \theta_{0}^{T}) a(s) \lambda_{s} Z(s)
   \]

2. **Separate the coefficients of \( a(s) \) on the left** and the terms on the right:

   Since both sides of the equation must balance, this implies a system of linear equations in terms of \(\theta_1\), \(\theta_2\), and \(\theta_0\). 

   Comparing coefficients:

   \[
   \theta_{1}^{T} a(s) P(s) = A(s) P(s)
   \]

   and:

   \[
   (\theta_{2}^{T} + \theta_{0}^{T}) a(s) \lambda_{s} Z(s) = -k_{\rho} Z(s) P_{\rho} (s)
   \]

3. **Solve for \(\theta_1\):**

   From the first equation:

   \[
   \theta_{1}^{T} a(s) = A(s)
   \]

   Therefore:

   \[
   \theta_1 = (a(s))^{-T} A(s) / P(s)
   \]

   if \( a(s) \) is invertible.

4. **Solve for \(\theta_2\) and \(\theta_0\):**

   From the second equation:

   \[
   (\theta_{2}^{T} + \theta_{0}^{T}) a(s) = -k_{\rho} P_{\rho}(s) / (\lambda_s Z(s))
   \]

   To solve explicitly, we may need further information or additional constraints about \(a(s)\), \(\lambda_s\), \(Z(s)\), and \(P_{\rho}(s)\).

Would you like a more detailed solution, or have any specific constraints or values to consider?

### Further Questions:
1. Do you have values or expressions for \(A(s)\), \(P(s)\), and \(a(s)\)?
2. Can you provide more context or assumptions about the functions involved?
3. Is \( a(s) \) guaranteed to be invertible?
4. Are \(\theta_1\), \(\theta_2\), and \(\theta_0\) vectors or scalars?
5. Are there specific boundary conditions or additional equations for these parameters?

### Tip:
When solving systems of equations involving parameters like \(\theta\), always check for linear independence of the equations to ensure a unique solution.

Solve the linear system of equations involving θ1, θ2, and θ0 with respect to functions A(s), P(s), and Z(s). This step-by-step guide covers key concepts in linear algebra, including system of equations and matrix inversion, suitable for advanced university-level students.

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