Four Friends Abhay, Bina, Chhaya And Devesh Were Asked To Simplify 4 AB+ 3(AB+ BA) - 4 BA, Where A And B Are Both Matrices Of Order 2 × 2. It Is Known That ABI And A-1 B. Their Answers Are Given As : # GAB + Loo Abhay 6 AB Bina : 7 AB-BA Chhaya: 8

Mar 8, 2025 by ADMIN 248 views

10. Four friends Abhay, Bina, Chhaya and Devesh were asked to simplify 4 AB+ 3(AB+ BA) - 4 BA, where A and B are both matrices of order 2 × 2.

Introduction

In this problem, we are given a mathematical expression involving matrices A and B of order 2 × 2. The expression is 4 AB+ 3(AB+ BA) - 4 BA. We need to simplify this expression using the properties of matrices. The friends Abhay, Bina, Chhaya, and Devesh have provided their answers, which are given as:

Abhay: 4 AB + 3 AB + 3 BA - 4 BA
Bina: 7 AB - BA
Chhaya: 8 AB
Devesh: ?

Step 1: Understanding the Properties of Matrices

To simplify the given expression, we need to understand the properties of matrices. Matrices are mathematical objects that can be added, subtracted, and multiplied. The addition and subtraction of matrices are done element-wise, while the multiplication of matrices is done using the dot product.

Step 2: Simplifying the Expression

Let's simplify the given expression step by step.

First, we can simplify the expression inside the parentheses:

3(AB+ BA) = 3 AB + 3 BA

Now, we can substitute this expression back into the original expression:

4 AB + 3 AB + 3 BA - 4 BA

Step 3: Combining Like Terms

We can combine like terms in the expression:

(4 AB + 3 AB) + (3 BA - 4 BA)

This simplifies to:

7 AB - BA

Step 4: Analyzing the Friends' Answers

Now, let's analyze the friends' answers:

Abhay: 4 AB + 3 AB + 3 BA - 4 BA
Bina: 7 AB - BA
Chhaya: 8 AB
Devesh: ?

Step 5: Finding the Correct Answer

From the previous steps, we have simplified the expression to:

7 AB - BA

This is the correct answer. Abhay's answer is close, but it has an extra term. Bina's answer is correct, but it can be simplified further. Chhaya's answer is incorrect.

Conclusion

In this problem, we simplified the expression 4 AB+ 3(AB+ BA) - 4 BA using the properties of matrices. We analyzed the friends' answers and found that the correct answer is 7 AB - BA.

Discussion

Abhay's answer is close, but it has an extra term. He needs to simplify the expression further.
Bina's answer is correct, but it can be simplified further. She needs to combine like terms.
Chhaya's answer is incorrect. He needs to simplify the expression using the properties of matrices.
Devesh's answer is not provided. He needs to simplify the expression using the properties of matrices.

Tips and Tricks

When simplifying expressions involving matrices, it's essential to understand the properties of matrices.
Combine like terms to simplify the expression.
Use the distributive property to simplify expressions involving parentheses.

Example Problems

Simplify the expression 2 AB + 3 BA - 4 AB.
Simplify the expression 4 AB + 2 BA - 3 BA.

Practice Problems

Simplify the expression 3 AB + 2 BA - 4 AB.
Simplify the expression 2 AB + 3 BA - 2 BA.

Solutions

Simplify the expression 3 AB + 2 BA - 4 AB: (3 AB + 2 BA) - 4 AB = 3 AB + 2 BA - 4 AB = (3 AB - 4 AB) + 2 BA = - AB + 2 BA
Simplify the expression 2 AB + 3 BA - 2 BA: (2 AB + 3 BA) - 2 BA = 2 AB + 3 BA - 2 BA = 2 AB + BA = 3 AB

Q&A: Simplifying Expressions Involving Matrices

Q: What are the properties of matrices that we need to understand to simplify expressions involving matrices?

A: To simplify expressions involving matrices, we need to understand the properties of matrices, such as addition, subtraction, and multiplication. We also need to understand the distributive property and how to combine like terms.

Q: How do we simplify expressions involving parentheses?

A: To simplify expressions involving parentheses, we need to use the distributive property. This means that we need to multiply the term outside the parentheses by each term inside the parentheses.

Q: What is the correct answer to the expression 4 AB+ 3(AB+ BA) - 4 BA?

A: The correct answer to the expression 4 AB+ 3(AB+ BA) - 4 BA is 7 AB - BA.

Q: Why is Abhay's answer incorrect?

A: Abhay's answer is incorrect because he has an extra term in his expression. He needs to simplify the expression further by combining like terms.

Q: Why is Bina's answer correct, but can be simplified further?

A: Bina's answer is correct, but it can be simplified further by combining like terms. She needs to simplify the expression 7 AB - BA to 6 AB.

Q: Why is Chhaya's answer incorrect?

A: Chhaya's answer is incorrect because he has not simplified the expression using the properties of matrices. He needs to use the distributive property and combine like terms to simplify the expression.

Q: What are some tips and tricks for simplifying expressions involving matrices?

A: Some tips and tricks for simplifying expressions involving matrices include:

Understanding the properties of matrices, such as addition, subtraction, and multiplication.
Using the distributive property to simplify expressions involving parentheses.
Combining like terms to simplify the expression.
Using the order of operations to simplify the expression.

Q: What are some example problems that we can use to practice simplifying expressions involving matrices?

A: Some example problems that we can use to practice simplifying expressions involving matrices include:

Simplify the expression 2 AB + 3 BA - 4 AB.
Simplify the expression 4 AB + 2 BA - 3 BA.

Q: What are some practice problems that we can use to practice simplifying expressions involving matrices?

A: Some practice problems that we can use to practice simplifying expressions involving matrices include:

Simplify the expression 3 AB + 2 BA - 4 AB.
Simplify the expression 2 AB + 3 BA - 2 BA.

Q: How do we know if our answer is correct?

A: We can check if our answer is correct by plugging it back into the original expression and simplifying it. If the simplified expression matches our answer, then we know that our answer is correct.

Q: What are some common mistakes that we can make when simplifying expressions involving matrices?

A: Some common mistakes that we can make when simplifying expressions involving matrices include:

Not using the distributive property to simplify expressions involving parentheses.
Not combining like terms to simplify the expression.
Not using the order of operations to simplify the expression.

Q: How can we avoid making these mistakes?

A: We can avoid making these mistakes by:

Understanding the properties of matrices, such as addition, subtraction, and multiplication.
Using the distributive property to simplify expressions involving parentheses.
Combining like terms to simplify the expression.
Using the order of operations to simplify the expression.

Q: What are some real-world applications of simplifying expressions involving matrices?

A: Some real-world applications of simplifying expressions involving matrices include:

Computer graphics: Matrices are used to perform transformations on images and 3D models.
Physics: Matrices are used to describe the motion of objects in space.
Engineering: Matrices are used to solve systems of linear equations.

Q: How can we use simplifying expressions involving matrices in our daily lives?

A: We can use simplifying expressions involving matrices in our daily lives by:

Using computer software to perform calculations involving matrices.
Solving systems of linear equations to optimize business decisions.
Using matrices to describe the motion of objects in space.

Q: What are some resources that we can use to learn more about simplifying expressions involving matrices?

A: Some resources that we can use to learn more about simplifying expressions involving matrices include:

Textbooks on linear algebra and matrix theory.
Online tutorials and videos on simplifying expressions involving matrices.
Practice problems and exercises on simplifying expressions involving matrices.

Q: How can we practice simplifying expressions involving matrices?

A: We can practice simplifying expressions involving matrices by:

Solving practice problems and exercises.
Using online resources and tutorials.
Working with a study group or tutor to practice simplifying expressions involving matrices.