The problem you've uploaded involves matrix multiplication. It shows two matrices:

$P = \begin{bmatrix} 1 \\ 2 \\ 3 \end{bmatrix} \quad \text{and} \quad Q = \begin{bmatrix} 1 & 2 & 3 \end{bmatrix}$

The task is to find the result of multiplying matrix $Q$ by matrix $P$, represented as $Q \cdot P$.

Explanation:

To multiply these matrices, we need to verify the dimensions of both matrices:

• Matrix $P$ is a column vector with dimensions $3 \times 1$.
• Matrix $Q$ is a row vector with dimensions $1 \times 3$.

The multiplication $Q \cdot P$ is defined because the number of columns in $Q$ (which is 3) matches the number of rows in $P$ (which is 3). The resulting matrix will have dimensions $3 \times 3$.

Step-by-step multiplication:

To multiply $Q \cdot P$, perform the following steps:

$Q \cdot P = \begin{bmatrix} 1 & 2 & 3 \end{bmatrix} \times \begin{bmatrix} 1 \\ 2 \\ 3 \end{bmatrix}$

Each element of the resulting matrix is found by multiplying each element of $Q$ by each element of $P$:

$Q \cdot P = \begin{bmatrix} 1 \cdot 1 & 1 \cdot 2 & 1 \cdot 3 \\ 2 \cdot 1 & 2 \cdot 2 & 2 \cdot 3 \\ 3 \cdot 1 & 3 \cdot 2 & 3 \cdot 3 \end{bmatrix}$

Thus, the result is:

$Q \cdot P = \begin{bmatrix} 1 & 2 & 3 \\ 2 & 4 & 6 \\ 3 & 6 & 9 \end{bmatrix}$

Conclusion:

The result of $Q \cdot P$ is:

$\begin{bmatrix} 1 & 2 & 3 \\ 2 & 4 & 6 \\ 3 & 6 & 9 \end{bmatrix}$

This corresponds to the third option in your question.

Would you like further clarification or any additional questions related to matrix operations? Here's a list of topics you might find interesting:

1. The properties of matrix multiplication.
2. How to multiply matrices with different dimensions.
3. The concept of the transpose of a matrix.
4. The role of identity matrices in matrix multiplication.
5. How to compute the determinant of a matrix.

Tip: When multiplying matrices, remember that the number of columns in the first matrix must match the number of rows in the second matrix for the multiplication to be defined!