https://file.aiquickdraw.com/mathbot/file/1728179021261xkfuwruj.jpg

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

1. **Matrix Multiplication (Top Row):**
   \[
   \begin{bmatrix}
   3 & 4 & 2 \\
   1 & 2 & 2
   \end{bmatrix}
   \times
   \begin{bmatrix}
   6.1 \\
   3.4 \\
   6.7
   \end{bmatrix}
   \]
   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):**
   \[
   8 \times \begin{bmatrix}
   3 & 4 & 2 \\
   1 & 2 & 2
   \end{bmatrix}
   +
   \begin{bmatrix}
   6.1 & 3.4 & 6.7
   \end{bmatrix}
   \]
   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):**
   \[
   \begin{bmatrix}
   3 & 4 & 2 \\
   1 & 2 & 2
   \end{bmatrix}
   +
   3 \times
   \begin{bmatrix}
   6.1 & 3.4 & 6.7
   \end{bmatrix}
   \]
   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):**
   \[
   2 \times \begin{bmatrix}
   3 & 4 & 2 \\
   1 & 2 & 2
   \end{bmatrix}
   \]
   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.

Which of these operations are defined? Select all that apply.

Analyze matrix operations involving multiplication, scalar multiplication, and addition. Understand when these operations are defined based on matrix dimensions and scalar rules. Suitable for students in Grades 10-12.

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