Matrix Addition Explained: A + B
Given Matrices and Their Sum: Let's dive into the world of matrices! We're going to perform matrix addition, a fundamental operation in linear algebra. The problem presents two matrices, A and B, and asks us to find their sum, denoted as A + B. This seemingly simple operation is the cornerstone of more complex matrix manipulations, so understanding it is key. We'll walk through the steps, making sure everything is clear and easy to follow. Plus, we'll make it fun, so you won't get bored. Remember those old math problems, well this is the improved version, just with matrices. Let's get started with a straightforward example and find the answer.
Understanding the Basics of Matrix Addition
What is Matrix Addition? Matrix addition, at its heart, is about combining two or more matrices. But there's a catch: you can only add matrices that have the same dimensions. Dimensions refer to the number of rows and columns a matrix possesses. If the matrices don't have the same dimensions, you can't perform the addition, it's as simple as that. Think of it like adding apples and oranges; they just don't mix! For A + B to be defined, A and B must be the same size. So, we'll go back to that matrix format that we know and love from the beginning of the problem, or at least, if you are like me, that we are just getting used to! For example, a 2x2 matrix can only be added to another 2x2 matrix. Now, let's explore what these dimensions are and how they affect the addition process.
Matrix Dimensions: Before adding matrices, we need to understand their dimensions. A matrix's dimensions are described as rows x columns. For instance, a matrix with 2 rows and 3 columns is a 2x3 matrix. In our case, both matrices A and B are 3x2 matrices. That means they each have 3 rows and 2 columns. Since they have the same dimensions, we can proceed with the addition. Matrix addition then becomes a walk in the park. Matrix dimensions play a crucial role in determining whether matrix addition is even possible. Now that we've established our matrices are compatible, let's get into how the actual addition takes place. Ready to dive in and work it out? Let's go. Matrix addition might seem daunting at first, but it's a piece of cake once you get the hang of it.
Step-by-Step Guide to Adding Matrices A and B
Adding Corresponding Elements: The core of matrix addition involves adding corresponding elements. This means you add the elements in the same positions in both matrices. For example, the element in the first row and first column of matrix A is added to the element in the first row and first column of matrix B. The same goes for every other element in the matrices. Adding matrices becomes a series of simple additions. This element-wise addition is the heart of matrix addition, and it's not as hard as it sounds! The structure of A + B will also be a 3x2 matrix. This is the result of adding two matrices that have the same dimensions. Let's write out the matrices again to make sure we are on the same page, it's always good to get things organized.
Let's calculate A + B: First, let's write out the matrices A and B so we can easily visualize the process.
Now, let's add the corresponding elements:
- Element (1,1): 3 (from A) + 3 (from B) = 6
- Element (1,2): -1 (from A) + 3 (from B) = 2
- Element (2,1): 2 (from A) + (-5) (from B) = -3
- Element (2,2): 0 (from A) + 4 (from B) = 4
- Element (3,1): -3 (from A) + (-4) (from B) = -7
- Element (3,2): 3 (from A) + 2 (from B) = 5
The Resulting Matrix: Finally, we compile these results into a new matrix, which represents A + B. The outcome is a 3x2 matrix, mirroring the dimensions of the original matrices.
Matrix Addition: The Answer and Beyond
The Solution: The sum of matrices A and B is: $A+B=\left[\begin{array}{cc}6 & 2 \ -3 & 4 \ -7 & 5\end{array}\right]$. This result is what we get when we add the corresponding elements of matrices A and B. This whole matrix addition thing might seem difficult but the logic is simple.
Conclusion: In summary, we've successfully added matrices A and B. We've covered the fundamental steps of matrix addition, ensuring you can now confidently add matrices of the same dimensions. Remember, matrix addition is a fundamental concept, so understanding this now will make more complicated matrix operations, such as matrix multiplication, a whole lot easier in the long run. Adding matrices is like doing a simple arithmetic problem but in a structured format. The principles of matrix addition are consistent no matter the size or values of the matrix as long as the dimensions are compatible. The beauty of matrix addition lies in its simplicity and elegance in handling data. Congrats, you've just conquered matrix addition!