The problem in the image asks about matrix operations and their validity given the dimensions of the matrices. Here's how we can evaluate each option:

• Matrix Dimensions:
  • $A$ and $B$ are both $3 \times 8$ matrices.
  • $C$ is a $7 \times 3$ matrix.

Now, let’s analyze each option:

Option A: $B + C$

• Matrix addition requires the matrices to have the same dimensions.
• $B$ is $3 \times 8$, and $C$ is $7 \times 3$, so they do not have matching dimensions.
• Not defined.

Option B: $B + A$

• Both $B$ and $A$ are $3 \times 8$ matrices.
• Since their dimensions match, matrix addition is defined.
• Defined.

Option C: $BC$

• For matrix multiplication, the number of columns of the first matrix must equal the number of rows of the second matrix.
• $B$ is $3 \times 8$ and $C$ is $7 \times 3$, so the number of columns of $B$ (8) does not match the number of rows of $C$ (7).
• Not defined.

Option D: $CB + 2A$

• First, evaluate $CB$:
  • $C$ is $7 \times 3$ and $B$ is $3 \times 8$.
  • The number of columns of $C$ (3) matches the number of rows of $B$ (3), so the matrix product $CB$ is defined and will have dimensions $7 \times 8$.
• Now, evaluate $2A$:
  • $A$ is $3 \times 8$. The scalar multiplication $2A$ will still be a $3 \times 8$ matrix.
  • However, you cannot add a $7 \times 8$ matrix (from $CB$) to a $3 \times 8$ matrix (from $2A$), as their dimensions are different.
• Not defined.

Option E: $CB$

• As previously discussed in Option D, $C$ is $7 \times 3$ and $B$ is $3 \times 8$.
• The matrix multiplication is defined and results in a $7 \times 8$ matrix.
• Defined.

Conclusion:

The valid (defined) options are B (B + A) and E (CB).

Let me know if you would like further details or have any questions!

Here are 5 questions to expand on the topic:

1. How can matrix multiplication be generalized for arbitrary dimensions?
2. What are the conditions required for matrix addition?
3. If matrix $A$ is $m \times n$ and matrix $B$ is $n \times p$, what will be the dimensions of $AB$?
4. How does scalar multiplication differ from matrix multiplication?
5. What happens when you multiply a matrix by its transpose?

Tip: Always check the dimensions of the matrices involved before performing operations to avoid undefined results.