Let us solve the given matrix problems step by step. The matrices provided are:

$A = \begin{pmatrix} 10 & 3 \\ -2 & 1 \end{pmatrix}, \quad B = \begin{pmatrix} 8 & 4 \\ -1 & -3 \end{pmatrix}, \quad C = \begin{pmatrix} 5 & -4 & 1 \\ 3 & -2 & 0 \end{pmatrix}.$

We need to calculate the following:

1. $B^{-1}$ (the inverse of $B$).
2. $3A - B$.
3. $(A + B)^T$ (the transpose of $A + B$).
4. $A + C$.

Let me calculate these one at a time. numpy as np

Define the matrices A, B, and C

A = np.array([[10, 3], [-2, 1]]) B = np.array([[8, 4], [-1, -3]]) C = np.array([[5, -4, 1], [3, -2, 0]])

1. Compute the inverse of B (B^-1)

B_inverse = np.linalg.inv(B)

2. Compute 3A - B

result_3A_minus_B = 3 * A - B

3. Compute (A + B)^T

result_A_plus_B_transpose = (A + B).T

Since A and C have incompatible dimensions for direct addition, we cannot compute A + C.

B_inverse, result_3A_minus_B, result_A_plus_B_transposeHere are the calculated results:

1. The inverse of $B$ ($B^{-1}$) is: $B^{-1} = \begin{pmatrix} 0.15 & 0.2 \\ -0.05 & -0.4 \end{pmatrix}$

2. $3A - B$ is: $3A - B = \begin{pmatrix} 22 & 5 \\ -5 & 6 \end{pmatrix}$

3. The transpose of $A + B$ ($(A + B)^T$) is: $(A + B)^T = \begin{pmatrix} 18 & -3 \\ 7 & -2 \end{pmatrix}$

Note: $A + C$ cannot be computed because the matrices $A$ (2x2) and $C$ (2x3) have incompatible dimensions for addition.

Let me know if you need further clarifications or additional details! Here are 5 follow-up questions to deepen your understanding:

1. What are the conditions for a matrix to have an inverse?
2. How is the transpose of a matrix defined?
3. What happens if two matrices of incompatible dimensions are added or multiplied?
4. Can you verify the inverse of $B$ by multiplying $B$ with $B^{-1}$?
5. What are the practical applications of matrix operations like these?

Tip: Always check the dimensions of matrices before performing operations like addition or multiplication!