The problem shown in the image asks for the result of matrix multiplication between two matrices, $A$ and $B$, where:

$A = \begin{pmatrix} 6 & 7 \\ 2 & 3 \end{pmatrix}, \quad B = \begin{pmatrix} -5 \\ 4 \end{pmatrix}$

Matrix multiplication between matrix $A$ (which is a 2x2 matrix) and matrix $B$ (which is a 2x1 matrix) is possible, and the resulting matrix will be a 2x1 matrix.

Step-by-Step Solution

To multiply $A$ and $B$, we compute the dot product of each row of matrix $A$ with matrix $B$:

$A \cdot B = \begin{pmatrix} 6 & 7 \\ 2 & 3 \end{pmatrix} \cdot \begin{pmatrix} -5 \\ 4 \end{pmatrix}$

First element of the resulting matrix:

$(6 \times -5) + (7 \times 4) = -30 + 28 = -2$

Second element of the resulting matrix:

$(2 \times -5) + (3 \times 4) = -10 + 12 = 2$

Result:

The result of multiplying matrices $A$ and $B$ is:

$\begin{pmatrix} -2 \\ 2 \end{pmatrix}$

This corresponds to the first option on the left in the given choices.

Would you like more details on matrix multiplication or further clarification?

Here are five related questions for practice:

1. What happens when you multiply two 2x2 matrices together?
2. How would the result change if matrix $B$ had different values?
3. Can you find the inverse of matrix $A$, if it exists?
4. What is the determinant of matrix $A$?
5. How does matrix multiplication differ from element-wise multiplication?

Tip: Always check the dimensions of matrices before multiplying to ensure that the operation is valid.