https://file.aiquickdraw.com/mathbot/file/1725471282167he6lcj0p.jpg

The image depicts a matrix multiplication problem. The matrices involved are:

\[
\begin{pmatrix}
4 & 2 \\
5 & 3 \\
-1 & 7
\end{pmatrix}
\cdot
\begin{pmatrix}
1 \\
-1
\end{pmatrix}
\]

Let's multiply these matrices.

The result of the multiplication will be a 3x1 matrix. Each element is calculated as follows:

1. **First Row:**
   \[
   4 \times 1 + 2 \times (-1) = 4 - 2 = 2
   \]

2. **Second Row:**
   \[
   5 \times 1 + 3 \times (-1) = 5 - 3 = 2
   \]

3. **Third Row:**
   \[
   -1 \times 1 + 7 \times (-1) = -1 - 7 = -8
   \]

So, the product of the matrices is:

\[
\begin{pmatrix}
2 \\
2 \\
-8
\end{pmatrix}
\]

Would you like further details or explanations?

Here are five additional questions related to this topic:

1. What is the general process for multiplying two matrices?
2. How can matrix multiplication be applied in solving systems of linear equations?
3. What properties does matrix multiplication have, such as commutativity or associativity?
4. How does matrix multiplication differ from scalar multiplication?
5. What are some real-world applications of matrix multiplication?

**Tip:** Always ensure the dimensions of the matrices are compatible for multiplication; the number of columns in the first matrix must equal the number of rows in the second matrix.

What is the result of multiplying the matrix [[4, 2], [5, 3], [-1, 7]] with the vector [1, -1]?

This page covers how to multiply a 3x2 matrix [[4, 2], [5, 3], [-1, 7]] by a 2x1 vector [1, -1], resulting in a 3x1 vector. The solution involves matrix multiplication techniques, perfect for students in Grades 9-12.

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