Matrix Multiplication: Find R(A) * R(B) For Matrices A And B

Oct 17, 2025 by ADMIN 61 views

Matrix Multiplication and Rank Calculation: A Detailed Guide

Hey guys! Let's dive into a cool math problem involving matrix multiplication and rank calculation. We'll break down the concepts, go through the solution step-by-step, and make sure everything is super clear. So, grab your coffee, and let's get started!

Understanding the Problem: Matrices A and B

Alright, let's start by understanding the problem. We're given two matrices, A and B. Matrix A is a 3x3 matrix, and matrix B is a 3x4 matrix. The question asks us to find the product of the ranks of these two matrices, denoted as r(A) * r(B).

Before we begin, remember that the rank of a matrix is the maximum number of linearly independent rows or columns in the matrix. It's a fundamental concept in linear algebra, and it tells us a lot about the matrix's properties and the solutions of related linear equations. So, to solve this problem, we need to determine the rank of each matrix individually and then multiply the results. This might seem a bit tricky at first, but trust me, it's totally manageable once you understand the steps involved. We will analyze the matrices and solve it step by step to find the answer. It is very important to get a good grip on the concept of rank. This will help you greatly as you move forward. Now that we've got the basics down, let's get started with finding the rank of matrix A. Let's make sure we understand the matrix A.

Matrix A Breakdown

Matrix A is defined as:

A = | -1  2  0 |
    |  3  2 -3 |
    |  4  0 -3 |

To find the rank of matrix A, we can use a method called Gaussian elimination. This process involves performing elementary row operations to transform the matrix into a row-echelon form. The number of non-zero rows in the row-echelon form is equal to the rank of the matrix. So, what exactly are we going to do with the rows? We'll perform operations like swapping rows, multiplying a row by a non-zero scalar, and adding a multiple of one row to another. The goal is to get as many zeros as possible below the leading non-zero elements in each row (also known as pivots). This will make it easier to see how many linearly independent rows there are. In simple terms, it's about simplifying the matrix without changing its essential properties, like its rank. After we simplify it, we can tell you how many rows are nonzero and that will be the rank of the matrix. Don't worry, we'll go through this step-by-step. The process is not that hard and it is easy to master. Are you ready?

Finding the Rank of Matrix A (r(A))

Let's apply Gaussian elimination to matrix A.

Step 1: Focus on the first column. We want to get zeros below the first element in the first row (-1). Multiply the first row by 3 and add it to the second row. Also, multiply the first row by 4 and add it to the third row.
```
A -> | -1  2  0 |
     |  0  8 -3 |
     |  0  8 -3 |
```
Step 2: Focus on the second column. Notice that the second and third rows are now identical. Subtract the second row from the third row.
```
A -> | -1  2  0 |
     |  0  8 -3 |
     |  0  0  0 |
```

Now, the matrix is in row-echelon form. We have two non-zero rows. Therefore, the rank of matrix A, denoted as r(A), is 2.

Analyzing Matrix B and Calculating its Rank

Alright, now that we've found the rank of matrix A, let's move on to matrix B. Matrix B is defined as a 3x4 matrix:

B = |  2  0 -1  3 |
    | -6  0  3 -9 |
    |  4  0 -2  6 |

Similar to matrix A, we'll use Gaussian elimination to find the rank of matrix B. We'll perform the same elementary row operations to get the matrix into row-echelon form. Remember, the rank will be the number of non-zero rows after we're done. Let's get started. Matrix B might look a little more complicated at first glance, but the process is exactly the same.

Calculating the Rank of Matrix B (r(B))

Let's apply Gaussian elimination to matrix B.

Step 1: Focus on the first column. We want to get zeros below the first element in the first row (2). Multiply the first row by 3 and add it to the second row. Also, multiply the first row by -2 and add it to the third row.
```
B -> |  2  0 -1  3 |
     |  0  0  0  0 |
     |  0  0  0  0 |
```

Now, the matrix is in row-echelon form. We have only one non-zero row. Therefore, the rank of matrix B, denoted as r(B), is 1.

Finding the Product of the Ranks: r(A) * r(B)

Great job, guys! We've successfully calculated the ranks of both matrices A and B. Now, the final step is to find the product of these ranks. We have:

r(A) = 2
r(B) = 1

Therefore, r(A) * r(B) = 2 * 1 = 2. This is a pretty straightforward calculation once we've found the ranks. Always remember to double-check your calculations. It's easy to make small mistakes, so take your time and make sure you're confident in your answers. So, our final result is 2. Now let's wrap this up with a summary of the whole process.

Conclusion: The Final Answer and Summary

To recap everything we've done, we started by understanding the concept of rank and its importance in linear algebra. Then, we were given two matrices, A and B, and we were asked to find the product of their ranks. We used Gaussian elimination to transform each matrix into its row-echelon form. This allowed us to determine the rank of each matrix easily. We found that the rank of matrix A is 2, and the rank of matrix B is 1. Finally, we multiplied these ranks together to get our final answer: r(A) * r(B) = 2. We used the properties of matrix multiplication and linear algebra to determine the solution. The correct answer is therefore option 4, which is 2.

Key Takeaways

Rank of a Matrix: Understand what the rank of a matrix represents (maximum number of linearly independent rows or columns).
Gaussian Elimination: Master the technique of Gaussian elimination for finding the rank.
Step-by-Step Approach: Break down complex problems into smaller, manageable steps.
Double-Check Your Work: Always review your calculations and steps to minimize errors.

I hope this detailed explanation was helpful! If you've got any more questions or want to practice some more problems, feel free to ask. Keep up the awesome work, and keep exploring the amazing world of linear algebra!