Let's analyze the given operations to determine which are defined.

1. Matrix Multiplication (Top Row): 3 & 4 & 2 \\ 1 & 2 & 2 \end{bmatrix} \times \begin{bmatrix} 6.1 \\ 3.4 \\ 6.7 \end{bmatrix}$$ The first matrix is $$2 \times 3$$ (2 rows and 3 columns), and the second matrix (a vector) is $$3 \times 1$$ (3 rows, 1 column). Matrix multiplication is possible when the number of columns of the first matrix matches the number of rows of the second matrix. Since this condition holds, this operation is **defined**.
2. Scalar Multiplication and Addition (Second Row): 3 & 4 & 2 \\ 1 & 2 & 2 \end{bmatrix} + \begin{bmatrix} 6.1 & 3.4 & 6.7 \end{bmatrix}$$ Scalar multiplication (multiplying a matrix by 8) is always defined. The result will be a $$2 \times 3$$ matrix. Addition requires both matrices to have the same dimensions. The matrix on the left is $$2 \times 3$$ while the matrix on the right is a $$1 \times 3$$ vector. Since their dimensions are not the same, **this operation is not defined**.
3. Matrix Addition with Scalar Multiplication (Third Row): 3 & 4 & 2 \\ 1 & 2 & 2 \end{bmatrix} + 3 \times \begin{bmatrix} 6.1 & 3.4 & 6.7 \end{bmatrix}$$ Scalar multiplication is defined, so the vector will be scaled by 3, giving a $$1 \times 3$$ vector. However, trying to add a $$2 \times 3$$ matrix with a $$1 \times 3$$ vector is not allowed because the dimensions must match. Therefore, **this operation is not defined**.
4. Scalar Multiplication (Bottom Row): 3 & 4 & 2 \\ 1 & 2 & 2 \end{bmatrix}$$ This is a straightforward scalar multiplication, which is always defined for matrices. Therefore, **this operation is defined**.

Conclusion:

• The first and last operations are defined.
• The second and third operations are not defined.

Would you like any further clarification or details on matrix operations? Here are 5 related questions you might find interesting:

1. How does matrix dimension affect addition and multiplication?
2. What are the conditions for matrix multiplication to be defined?
3. Can a scalar be multiplied with any matrix, regardless of dimensions?
4. How do we handle the addition of matrices of different sizes?
5. What are some real-world applications of matrix operations?

Tip: Always check the dimensions of matrices before performing operations like multiplication or addition to ensure they are defined.