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We are given a vector \(\vec{v} = \begin{bmatrix} 2 \\ -1 \\ 6 \end{bmatrix}\) and a matrix \(A = \begin{bmatrix} 3 & 2 \\ 2 & 4 \\ 5 & -1 \end{bmatrix}\). The question asks for the value of \( A\vec{v} \), which means we need to multiply matrix \(A\) by vector \(\vec{v}\).

Let's perform the matrix-vector multiplication.

\[
A\vec{v} = \begin{bmatrix} 3 & 2 \\ 2 & 4 \\ 5 & -1 \end{bmatrix} \times \begin{bmatrix} 2 \\ -1 \\ 6 \end{bmatrix}
\]

Multiplying each row of \(A\) with the corresponding elements of \(\vec{v}\):

1. First row: \( (3 \times 2) + (2 \times -1) = 6 - 2 = 4 \)
2. Second row: \( (2 \times 2) + (4 \times -1) = 4 - 4 = 0 \)
3. Third row: \( (5 \times 2) + (-1 \times -1) = 10 + 1 = 11 \)

So, the result of \( A\vec{v} \) is:

\[
A\vec{v} = \begin{bmatrix} 4 \\ 0 \\ 11 \end{bmatrix}
\]

Thus, \( A\vec{v} = \begin{bmatrix} 4 \\ 0 \\ 11 \end{bmatrix} \).

Would you like further details on the steps or have any questions?

Here are some related questions:
1. What are the conditions for matrix-vector multiplication to be defined?
2. How do you calculate the determinant of a matrix?
3. What are the applications of matrix-vector multiplication in real life?
4. What happens if we switch the order of multiplication (i.e., multiplying a vector by a matrix on the left)?
5. How would we interpret this result geometrically?

**Tip:** Always check the dimensions of the matrix and the vector to ensure multiplication is valid.

For \(\vec{v} = \begin{bmatrix} 2 \\ -1 \\ 6 \end{bmatrix}\) and matrix \(A = \begin{bmatrix} 3 & 2 \\ 2 & 4 \\ 5 & -1 \end{bmatrix}\), what is the value of \(A\vec{v}\)?

Learn how to solve matrix-vector multiplication for A\vec{v} with the given matrix A and vector \(\vec{v}\). This problem includes key linear algebra concepts, such as matrix and vector multiplication, and is suitable for students in Grades 10-12.

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