Let A and B be m × n matrices. Simplify the expression
2[9(A − B) + 7(2B − A)] − 2[3(2B + A) − 2(A + 3B) − 5(A + B)]


We are tasked with simplifying the following matrix expression:

\[
2[9(A − B) + 7(2B − A)] − 2[3(2B + A) − 2(A + 3B) − 5(A + B)]
\]

### Step 1: Simplify inside the brackets

#### First bracket: \( 9(A - B) + 7(2B - A) \)

1. Distribute \( 9 \) and \( 7 \) inside their respective terms:
   \[
   9(A - B) = 9A - 9B
   \]
   \[
   7(2B - A) = 14B - 7A
   \]
2. Combine like terms:
   \[
   9A - 9B + 14B - 7A = (9A - 7A) + (-9B + 14B) = 2A + 5B
   \]
So, the simplified form of the first bracket is \( 2A + 5B \).

#### Second bracket: \( 3(2B + A) - 2(A + 3B) - 5(A + B) \)

1. Distribute \( 3 \), \( -2 \), and \( -5 \):
   \[
   3(2B + A) = 6B + 3A
   \]
   \[
   -2(A + 3B) = -2A - 6B
   \]
   \[
   -5(A + B) = -5A - 5B
   \]
2. Combine like terms:
   \[
   (6B + 3A) - (2A + 6B) - (5A + 5B)
   \]
   Group the terms:
   \[
   (3A - 2A - 5A) + (6B - 6B - 5B) = -4A - 5B
   \]
So, the simplified form of the second bracket is \( -4A - 5B \).

### Step 2: Apply the factors outside the brackets

Now we return to the full expression:
\[
2[2A + 5B] - 2[-4A - 5B]
\]

1. Distribute the factor of 2 to both expressions:
   \[
   2(2A + 5B) = 4A + 10B
   \]
   \[
   2(-4A - 5B) = -8A - 10B
   \]

### Step 3: Combine the results

Now, combine the two simplified expressions:
\[
(4A + 10B) - (-8A - 10B) = 4A + 10B + 8A + 10B
\]
Simplify:
\[
(4A + 8A) + (10B + 10B) = 12A + 20B
\]

### Final Answer:
The simplified expression is:
\[
12A + 20B
\]

Would you like further details on any step?

---

Here are 5 questions that expand on the topic:

1. How would the result change if the dimensions of matrices \( A \) and \( B \) were not the same?
2. Can matrix expressions like this be simplified differently when multiplied by scalar values?
3. What would happen if \( A \) and \( B \) were diagonal matrices? Would simplifications be easier?
4. How would the process change if transposes of \( A \) or \( B \) were involved?
5. Can the distributive property in matrix algebra be applied when working with non-commutative operations like matrix multiplication?

**Tip:** Always double-check your distribution steps when simplifying expressions involving matrices, as errors in signs or coefficients can lead to incorrect results.

Let A and B be m × n matrices. Simplify the expression 2[9(A − B) + 7(2B − A)] − 2[3(2B + A) − 2(A + 3B) − 5(A + B)]

Learn how to simplify a matrix expression using the distributive property and combining like terms. This detailed solution walks through each step of the simplification process for the expression 2[9(A − B) + 7(2B − A)] − 2[3(2B + A) − 2(A + 3B) − 5(A + B)], ideal for advanced high school or college students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation