The problem in the image appears to involve multiplying a vector by a matrix. The expression is:

$(2 \; -1) \begin{pmatrix} 1 & 2 & 3 \\ 1 & -1 & 4 \end{pmatrix}$

We have a row vector $(2, -1)$ and a matrix:

$\begin{pmatrix} 1 & 2 & 3 \\ 1 & -1 & 4 \end{pmatrix}.$

To find the result, we perform matrix multiplication between the row vector and the matrix. The operation is defined as follows:

$(2, -1) \times \begin{pmatrix} 1 & 2 & 3 \\ 1 & -1 & 4 \end{pmatrix} = (2 \cdot 1 + (-1) \cdot 1, \; 2 \cdot 2 + (-1) \cdot (-1), \; 2 \cdot 3 + (-1) \cdot 4).$

Let's compute each term step-by-step:

1. First element: $2 \cdot 1 + (-1) \cdot 1 = 2 - 1 = 1$
2. Second element: $2 \cdot 2 + (-1) \cdot (-1) = 4 + 1 = 5$
3. Third element: $2 \cdot 3 + (-1) \cdot 4 = 6 - 4 = 2$

Thus, the resulting vector is:

$(1, 5, 2).$

Final Answer:

$(2, -1) \times \begin{pmatrix} 1 & 2 & 3 \\ 1 & -1 & 4 \end{pmatrix} = (1, 5, 2).$

Would you like a detailed explanation of matrix multiplication or have any other questions?

Here are 5 related questions you might find interesting:

1. How do you multiply a column vector by a matrix?
2. What are the properties of matrix multiplication?
3. How is matrix multiplication used in linear transformations?
4. Can a matrix be multiplied by another matrix of any size?
5. What is the determinant of a 2x2 matrix?

Tip: Remember that matrix multiplication is not commutative; $AB \neq BA$ in general!