The document is a mid-term exam for a course titled "MENG 501 Computational Mathematics for Engineers" from Camarines Sur Polytechnic Colleges. Here is a breakdown of the exam questions:

1. Matrix Commutation Problem:

   • Two matrices are given: B = \begin{pmatrix} 1 & 2 \\ 1 & 0 \end{pmatrix}$$ The task is to find values of $$a$$ and $$b$$ such that $$AB = BA$$.

2. Matrix Inversion:

   • You are asked to find the inverse of the matrix: $A = \begin{pmatrix} 1 & 2 & 3 \\ 2 & 5 & 3 \\ 1 & 0 & 8 \end{pmatrix}$ using two different methods.

3. Eigenvalues and Eigenvectors:

   • Find the eigenvalues and eigenvectors of the matrix: $B = \begin{pmatrix} 3 & 1 & 0 \\ 0 & 3 & 1 \\ 3 & -7 & 8 \end{pmatrix}$

4. Mechanical and Electrical Engineering Problem:

   • Depending on your specialization:
     • For Electrical Engineering: Derive and write the system of equations for a given electric circuit in the form $A x = b$.
     • For Mechanical Engineering: Derive the equations for a compound-mass spring system and write them in the form $\dot{x} + Ax = 0$.

Would you like assistance with solving any specific problem from this exam?

Here are some follow-up questions that could expand your understanding:

1. Would you like to see the detailed solution to the matrix commutation problem?
2. Are you interested in learning both methods for finding a matrix inverse?
3. Should we go through the step-by-step process for calculating eigenvalues and eigenvectors?
4. Would you prefer a deeper explanation of how to set up matrix equations in electrical circuits?
5. Are you looking for help deriving the equations for the mechanical system?

Tip: When solving for eigenvalues, always check if the determinant of $\lambda I - A = 0$ simplifies your calculations!