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Calcula el producto de las matrises una por una paso por paso 

Let's proceed step by step with the matrix multiplication, as depicted in the images. 

### Matrix A (from the first image):
\[
\begin{bmatrix}
2 & 3 & 4 \\
-5 & 1 & 2 \\
4 & -7 & 2
\end{bmatrix}
\]

### Matrix X (the column vector):
\[
\begin{bmatrix}
x_1 \\
x_2 \\
x_3
\end{bmatrix}
\]

### Step-by-Step Matrix Multiplication
We will compute the matrix product \( A \times X \), where each element in the resulting vector is computed by multiplying the corresponding row of matrix \( A \) by the column vector \( X \).

#### First row of the product:
\[
(2 \times x_1) + (3 \times x_2) + (4 \times x_3)
\]
This results in:
\[
2x_1 + 3x_2 + 4x_3
\]

#### Second row of the product:
\[
(-5 \times x_1) + (1 \times x_2) + (2 \times x_3)
\]
This results in:
\[
-5x_1 + x_2 + 2x_3
\]

#### Third row of the product:
\[
(4 \times x_1) + (-7 \times x_2) + (2 \times x_3)
\]
This results in:
\[
4x_1 - 7x_2 + 2x_3
\]

### Resulting Vector
Thus, the final product is the following column vector:
\[
\begin{bmatrix}
2x_1 + 3x_2 + 4x_3 \\
-5x_1 + x_2 + 2x_3 \\
4x_1 - 7x_2 + 2x_3
\end{bmatrix}
\]

Would you like further details, or do you have any questions?

Here are 5 related questions:
1. How would this process change if matrix \( A \) was not a square matrix?
2. What is the geometric interpretation of multiplying a matrix by a vector?
3. Can you solve for \( x_1, x_2, x_3 \) given a specific output vector?
4. How does matrix inversion relate to solving systems of equations?
5. What happens if the matrix \( A \) has a determinant of 0?

**Tip:** When performing matrix multiplication, ensure that the number of columns in the first matrix equals the number of rows in the second matrix.

Calcula el producto de las matrises una por una paso por paso

Learn how to multiply a 3x3 matrix by a column vector with detailed step-by-step instructions. The problem covers matrix multiplication rules and linear algebra, suitable for students in Grades 9-12.

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