https://r2.erweima.ai/mathbot/image/1726506485005ghk2kv5h.jpg

The image shows a mathematical expression that involves determining a sum of two matrix products. Here's a breakdown of the given expression:

\[
(-4) \cdot \begin{pmatrix} 2 \\ -5 \\ 3 \end{pmatrix} + (-4) \cdot \begin{pmatrix} 4 \\ 4 \\ -5 \end{pmatrix}
\]

### Step 1: Distribute the scalar \(-4\) to each vector.

- For the first vector:
  \[
  (-4) \cdot \begin{pmatrix} 2 \\ -5 \\ 3 \end{pmatrix} = \begin{pmatrix} -4 \cdot 2 \\ -4 \cdot (-5) \\ -4 \cdot 3 \end{pmatrix} = \begin{pmatrix} -8 \\ 20 \\ -12 \end{pmatrix}
  \]

- For the second vector:
  \[
  (-4) \cdot \begin{pmatrix} 4 \\ 4 \\ -5 \end{pmatrix} = \begin{pmatrix} -4 \cdot 4 \\ -4 \cdot 4 \\ -4 \cdot (-5) \end{pmatrix} = \begin{pmatrix} -16 \\ -16 \\ 20 \end{pmatrix}
  \]

### Step 2: Add the resulting vectors.

\[
\begin{pmatrix} -8 \\ 20 \\ -12 \end{pmatrix} + \begin{pmatrix} -16 \\ -16 \\ 20 \end{pmatrix} = \begin{pmatrix} -8 + (-16) \\ 20 + (-16) \\ -12 + 20 \end{pmatrix} = \begin{pmatrix} -24 \\ 4 \\ 8 \end{pmatrix}
\]

Thus, the final result of the sum is:

\[
\boxed{\begin{pmatrix} -24 \\ 4 \\ 8 \end{pmatrix}}
\]

Would you like more details on any step?

### Related Questions:
1. How do you multiply a scalar by a vector in linear algebra?
2. What is the general method for adding two vectors?
3. How can vector scaling be applied in physics problems?
4. What are the properties of scalar multiplication in matrix operations?
5. How would this problem change if the scalar was positive instead of negative?

**Tip:** When performing vector operations, always apply scalar multiplication before vector addition or subtraction.

Determine the following sum: (-4) * [2; -5; 3] + (-4) * [4; 4; -5]

Learn how to solve a vector sum that involves scalar multiplication, focusing on operations with vectors [2; -5; 3] and [4; 4; -5]. This problem explores core concepts such as scalar multiplication and vector addition, suitable for students in grades 10-12 or early university.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation