The mid-term exam you provided for the course "MENG 501 Computational Mathematics for Engineers" includes the following problems:

1. Matrix Commutation Problem (25 Points):

   • Given matrices $A = \begin{pmatrix} a & b \\ 1 & 2 \end{pmatrix}$ and $B = \begin{pmatrix} 1 & 2 \\ 1 & 0 \end{pmatrix}$, where $a$ and $b$ are real parameters, find all values of $a$ and $b$ such that $AB = BA$.

2. Matrix Inversion (25 Points):

   • 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 (25 Points):

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

4. Engineering-Specific Problems (25 Points Each):

   • For Electrical and Electronics Engineering: Derive the equations for a simple electric circuit and represent them in matrix form $A x = b$.
   • For Mechanical Engineering: Derive the equations for a compound-mass spring system and express them in the form $x' + A x = 0$.

Would you like detailed solutions to any specific problem? If so, please specify which one.

Related Questions:

1. How can we determine if two matrices commute in general?
2. What are the common methods for finding the inverse of a matrix?
3. What is the process for calculating eigenvalues and eigenvectors of a 3x3 matrix?
4. How are electrical circuit equations typically formulated in matrix form?
5. What are the key differences between systems of differential equations in electrical and mechanical engineering?

Tip: When finding the inverse of a matrix, always check if the determinant is non-zero first. This ensures that the matrix is invertible.