https://file.aiquickdraw.com/mathbot/file/1727057731966hk74f174.jpg

Let's analyze the matrix-vector product shown in the image.

The matrix is:

\[
\begin{pmatrix}
1 & 2 & 1 \\
-1 & 1 & 0
\end{pmatrix}
\]

and the vector is:

\[
\begin{pmatrix}
1 \\
2 \\
3
\end{pmatrix}
\]

### Matrix Dimensions:
- The matrix is a \(2 \times 3\) matrix (2 rows, 3 columns).
- The vector is a \(3 \times 1\) column vector (3 rows, 1 column).

### Matrix-Vector Multiplication:
The multiplication between a \(2 \times 3\) matrix and a \(3 \times 1\) vector is valid since the number of columns in the matrix (3) matches the number of rows in the vector (3). The result will be a \(2 \times 1\) vector.

Let's compute the product step by step:

\[
\begin{pmatrix}
1 & 2 & 1 \\
-1 & 1 & 0
\end{pmatrix}
\begin{pmatrix}
1 \\
2 \\
3
\end{pmatrix}
=
\begin{pmatrix}
1(1) + 2(2) + 1(3) \\
-1(1) + 1(2) + 0(3)
\end{pmatrix}
=
\begin{pmatrix}
1 + 4 + 3 \\
-1 + 2 + 0
\end{pmatrix}
=
\begin{pmatrix}
8 \\
1
\end{pmatrix}
\]

### Conclusion:
The result is a vector in \( \mathbb{R}^2 \). Therefore, \( n = 2 \), which means the correct answer is **n = 2**.

Would you like more details or have any other questions?

Here are 5 related questions:
1. How do we multiply matrices with vectors in general?
2. What are the properties of matrix-vector multiplication?
3. What happens if the dimensions don't match in matrix-vector multiplication?
4. What is the geometric interpretation of matrix-vector multiplication?
5. How can matrix multiplication be extended to larger matrices?

**Tip:** Always check the dimensions of matrices and vectors before performing any multiplication to ensure the operation is valid.

The matrix-vector product 
\[
\begin{pmatrix}
1 & 2 & 1 \\
-1 & 1 & 0
\end{pmatrix}
\]
with the vector 
\[
\begin{pmatrix}
1 \\
2 \\
3
\end{pmatrix}
\] 
is a vector in \( \mathbb{R}^n \), where?

Understand the matrix-vector multiplication process with this example involving a 2x3 matrix and a 3x1 vector. Learn step-by-step how to verify the dimensions, multiply, and determine that the result is a vector in R^2. Suitable for students in Grades 10-12.

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