Matrix Addition: Finding A + B

Hey guys! Let's dive into the world of matrices and figure out how to add them. We've got two matrices, $A$ and $B$, and our mission is to find their sum, denoted as $A + B$. It's super straightforward, and I'll walk you through it step-by-step. Get ready to flex those math muscles! We'll explore the given matrices, understanding what they look like and how they're structured. Then, we'll apply the matrix addition rules, ensuring we match corresponding elements to calculate the new matrix, the sum of $A$ and $B$. Don't worry, it's not as scary as it sounds. Matrix addition is a fundamental concept in linear algebra, and understanding it will open doors to more complex mathematical operations. This process is used across various fields, including computer graphics, data science, and physics, so it is a good idea to have a grip on this. So, let's go!

Understanding the Matrices: $A$ and $B$

First off, let's take a closer look at our matrices. We have matrix $A$ and matrix $B$, each filled with numbers arranged in rows and columns. Think of them like grids of numbers. Matrix $A$ is given as: $A = \begin{pmatrix} 5 & 1 \\ 8 & -1 \\ -2 & 6 \\ \end{pmatrix}$. This is a 3x2 matrix, which means it has three rows and two columns. Each number within the matrix is called an element, and its position is defined by its row and column. For example, the element in the first row and first column of matrix $A$ is 5. Similarly, in the second row and second column of matrix $A$ is -1. Now, let's check out matrix $B$: $B = \begin{pmatrix} 6 & 2 \\ 9 & 7 \\ 8 & 4 \\ \end{pmatrix}$. Matrix $B$ is also a 3x2 matrix, matching the dimensions of matrix $A$. This is important because, in matrix addition, the matrices must have the same dimensions. If the dimensions don't match, you can't add them. This condition ensures that we can pair up the corresponding elements correctly. Now that we know what the matrices look like and understand their dimensions, we can move on to the actual addition process.

Before we jump into the addition, let's recap on the basics. Remember, matrices are arrays of numbers, organized into rows and columns. They're fundamental to representing and manipulating linear transformations. The size of a matrix (e.g., 3x2) is crucial; it tells us how many rows and columns it has. The elements within a matrix are the individual numbers, and their position matters. Knowing this will help us do the matrix addition, making sure that we perform all the steps correctly. Let's start with the addition process now!

Performing the Matrix Addition: $A + B$

Now for the fun part: adding the matrices! The rule for matrix addition is simple: you add the corresponding elements of the matrices. Since both $A$ and $B$ are 3x2 matrices, we can proceed. To find $A + B$, we add each element of matrix $A$ to the corresponding element of matrix $B$. Let's break it down step-by-step to avoid any confusion. First, we add the elements in the first row, first column: 5 (from matrix $A$) + 6 (from matrix $B$) = 11. Next, we add the elements in the first row, second column: 1 (from matrix $A$) + 2 (from matrix $B$) = 3. Moving on to the second row, first column: 8 (from matrix $A$) + 9 (from matrix $B$) = 17. For the second row, second column: -1 (from matrix $A$) + 7 (from matrix $B$) = 6. Now, for the third row, first column: -2 (from matrix $A$) + 8 (from matrix $B$) = 6. Finally, the third row, second column: 6 (from matrix $A$) + 4 (from matrix $B$) = 10. By performing these additions, we are essentially creating a new matrix that represents the sum of the two original matrices.

So, putting it all together, the matrix $A + B$ looks like this: $A + B = \begin{pmatrix} 11 & 3 \\ 17 & 6 \\ 6 & 10 \\ \end{pmatrix}$. And there you have it! We have successfully added matrices $A$ and $B$. Each element in the resulting matrix is the sum of the corresponding elements from the original matrices. See, that wasn't too bad, right? We have now computed the sum of the two matrices, a fundamental operation in linear algebra. This process is key when you're working with data transformations or solving systems of equations. Remember, the key is matching the corresponding elements and making sure your matrices have the same dimensions. This concept of adding matrices is the basis for more complex operations, making it essential to have a solid grasp of this process.

The Final Answer

So, to recap, the sum of matrices $A$ and $B$ is $A + B = \begin{pmatrix} 11 & 3 \\ 17 & 6 \\ 6 & 10 \\ \end{pmatrix}$. Congratulations, guys, you did it! We successfully added two matrices. Matrix addition is a cornerstone of linear algebra, and understanding how to perform it is essential for tackling more advanced concepts. Remember the simple rules: matching dimensions and adding corresponding elements. This skill comes into play in various fields, from computer graphics to data analysis, enabling you to manipulate and understand complex datasets. Keep practicing, and you'll become a matrix master in no time! Keep in mind that matrices play a huge role in mathematics and the sciences. They are used to represent data, transformations, and systems of equations. Understanding the basics, like matrix addition, is a great foundation to tackle more complex topics like matrix multiplication, finding determinants, and solving systems of linear equations. Now, you should be able to perform matrix addition, always ensuring that the matrices have the same dimensions before you start. Understanding this is a major step. Well done!